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Algebra, quadratics and simultaneous equations

## Quadratics

Quadratics are expressions or equations where the highest power of the variable (usually x) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Solving Quadratic Equations

There are several methods to solve quadratic equations:

  • Factorising: If the quadratic can be factorised, set each factor to zero to find the solutions (roots). For example, x² - 5x + 6 = 0 factors to (x-2)(x-3) = 0, so x=2 or x=3.
  • Quadratic Formula: This method always works. The formula is x = [-b ± √(b² - 4ac)] / 2a. Make sure the equation is in the form ax² + bx + c = 0 before identifying a, b, and c.
  • Completing the Square: This method transforms ax² + bx + c into the form a(x+p)² + q. It's useful for finding the turning point of a quadratic graph and for solving equations. For x² + bx + c, it becomes (x + b/2)² - (b/2)² + c.

The Discriminant

The discriminant is the part of the quadratic formula under the square root: b² - 4ac. It tells you about the nature of the roots:

  • If b² - 4ac > 0, there are two distinct real roots.
  • If b² - 4ac = 0, there is one repeated real root.
  • If b² - 4ac < 0, there are no real roots.

## Simultaneous Equations

Solving simultaneous equations means finding the values of the variables that satisfy *all* equations at the same time.

Linear Simultaneous Equations

These involve two or more linear equations. The main methods are:

  • Elimination: Multiply one or both equations to make the coefficients of one variable the same (or additive inverses), then add or subtract the equations to eliminate that variable.
  • Substitution: Rearrange one equation to express one variable in terms of the other, then substitute this expression into the second equation.

Linear and Quadratic Simultaneous Equations

These involve one linear equation and one quadratic equation. Always use the substitution method:

1. Rearrange the linear equation to make one variable the subject (e.g., y = mx + c).

2. Substitute this expression into the quadratic equation.

3. This will result in a quadratic equation in a single variable. Solve this quadratic equation (using factorising or the quadratic formula).

4. Substitute the found values back into the linear equation to find the corresponding values of the other variable.

The solutions represent the points of intersection between the line and the curve (parabola). There can be two solutions, one solution (tangent), or no solutions.

  • A quadratic equation is in the form ax² + bx + c = 0.
  • The quadratic formula x = [-b ± √(b² - 4ac)] / 2a always solves quadratic equations.
  • Completing the square (x + b/2)² - (b/2)² + c helps find the turning point of a parabola.
  • The discriminant b² - 4ac determines the number of real roots: >0 (two), =0 (one), <0 (none).
  • Linear simultaneous equations can be solved by elimination or substitution.
  • Solving linear-quadratic simultaneous equations always involves substituting the linear into the quadratic.
  • Solutions to simultaneous equations are the points where their graphs intersect.
What is the general form of a quadratic equation?
ax² + bx + c = 0
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State the quadratic formula.
x = [-b ± √(b² - 4ac)] / 2a
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What does the discriminant (b² - 4ac) tell you if it is less than 0?
There are no real roots.
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How do you solve linear-quadratic simultaneous equations?
Rearrange the linear equation to make one variable the subject, then substitute it into the quadratic equation.
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What method is best for solving 3x + 2y = 7 and 5x - 2y = 1?
Elimination (add the equations to eliminate y).
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What is the completed square form of x² + 6x + 5?
(x + 3)² - 4
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If (x-4)(x+1) = 0, what are the solutions for x?
x = 4 or x = -1
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Surds, indices and standard form

## Surds

Surds are irrational numbers that cannot be expressed as a simple fraction and contain a root symbol (e.g., `sqrt(2)`). They are exact values.

  • Simplifying Surds: Look for the largest perfect square factor. The rule `sqrt(ab) = sqrt(a) * sqrt(b)` is key. E.g., `sqrt(48) = sqrt(16 * 3) = 4sqrt(3)`.
  • Adding/Subtracting Surds: Only like surds (those with the same surd part) can be added or subtracted. Simplify first if necessary. E.g., `2sqrt(5) + 3sqrt(5) = 5sqrt(5)`.
  • Multiplying Surds: `sqrt(a) * sqrt(b) = sqrt(ab)`. E.g., `sqrt(3) * sqrt(7) = sqrt(21)`. When multiplying expressions like `(a + sqrt(b))(c + sqrt(d))`, use FOIL or grid method.
  • Rationalising the Denominator: This process removes the surd from the denominator.
  • If the denominator is `sqrt(a)`, multiply the numerator and denominator by `sqrt(a)`. E.g., `1/sqrt(3) = (1 * sqrt(3))/(sqrt(3) * sqrt(3)) = sqrt(3)/3`.
  • If the denominator is `a + sqrt(b)` or `a - sqrt(b)`, multiply by its conjugate. The conjugate of `a + sqrt(b)` is `a - sqrt(b)`. E.g., `1/(2 + sqrt(3)) = (1 * (2 - sqrt(3)))/((2 + sqrt(3))(2 - sqrt(3))) = (2 - sqrt(3))/(4 - 3) = 2 - sqrt(3)`.

## Indices

Indices (or powers) indicate how many times a base number is multiplied by itself.

  • Laws of Indices:
  • `a^m * a^n = a^(m+n)` (Multiplication)
  • `a^m / a^n = a^(m-n)` (Division)
  • `(a^m)^n = a^(mn)` (Power of a power)
  • `a^0 = 1` (Any non-zero number to the power of zero)
  • Negative Indices: `a^(-n) = 1/a^n`. E.g., `5^(-2) = 1/5^2 = 1/25`.
  • Fractional Indices:
  • `a^(1/n) = n-th root of a`. E.g., `16^(1/2) = sqrt(16) = 4`.
  • `a^(m/n) = (n-th root of a)^m` or `n-th root of (a^m)`. E.g., `27^(2/3) = (cube root of 27)^2 = 3^2 = 9`.

## Standard Form

Standard form is a concise way to write very large or very small numbers. It is written as `A x 10^n`, where `1 <= A < 10` and `n` is an integer.

  • Writing in Standard Form:
  • Move the decimal point until `A` is between 1 and 10 (inclusive of 1, exclusive of 10).
  • `n` is the number of places the decimal point was moved. `n` is positive for large numbers, negative for small numbers.
  • E.g., `45000 = 4.5 x 10^4`. `0.000072 = 7.2 x 10^(-5)`.
  • Calculations:
  • Multiplication/Division: Multiply/divide the `A` parts and add/subtract the powers of 10. Adjust `A` if it falls outside the `1 <= A < 10` range.
  • E.g., `(2 x 10^3) * (3 x 10^4) = (2*3) x 10^(3+4) = 6 x 10^7`.
  • Addition/Subtraction: Make the powers of 10 the same before adding/subtracting, or convert to ordinary numbers. Then convert the result back to standard form if needed.
  • E.g., `3 x 10^5 + 2 x 10^4 = 3 x 10^5 + 0.2 x 10^5 = 3.2 x 10^5`.
  • A surd is an irrational number containing a root symbol, often simplified by finding perfect square factors.
  • Rationalise a denominator with `sqrt(a)` by multiplying by `sqrt(a)/sqrt(a)`, or by its conjugate if it's `a ± sqrt(b)`.
  • The laws of indices are crucial: `a^m * a^n = a^(m+n)`, `a^m / a^n = a^(m-n)`, and `(a^m)^n = a^(mn)`.
  • A negative index means the reciprocal: `a^(-n) = 1/a^n`.
  • A fractional index `a^(m/n)` means the `n`-th root of `a` raised to the power `m`.
  • Any non-zero number raised to the power of 0 is 1 (`a^0 = 1`).
  • Standard form is `A x 10^n`, where `1 <= A < 10` and `n` is an integer.
  • When multiplying or dividing in standard form, multiply/divide the `A` parts and add/subtract the powers of 10, then adjust `A` if necessary.
Simplify `sqrt(72)`
`6sqrt(2)` (`sqrt(36 * 2) = 6sqrt(2)`)
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Rationalise `3/sqrt(5)`
`3sqrt(5)/5`
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What is `(x^4)^3`?
`x^12`
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Express `y^(-5)` using a positive index.
`1/y^5`
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What is `64^(2/3)`?
`16` (`(cube root of 64)^2 = 4^2 = 16`)
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Write `0.00000091` in standard form.
`9.1 x 10^(-7)`
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Calculate `(3 x 10^4) * (5 x 10^2)` in standard form.
`1.5 x 10^7` (`15 x 10^6 = 1.5 x 10^7`)
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Simplify `sqrt(20) + sqrt(45)`
`5sqrt(5)` (`2sqrt(5) + 3sqrt(5) = 5sqrt(5)`)
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Trigonometry and Pythagoras

## Pythagoras' Theorem

Pythagoras' Theorem applies to right-angled triangles only. It states that the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides. If the sides are `a`, `b`, and `c` (where `c` is the hypotenuse), then `a² + b² = c²`.

To find a shorter side, rearrange to `a² = c² - b²`. This theorem is crucial for finding missing lengths in 2D and 3D shapes, such as diagonals of cuboids.

## Right-Angled Trigonometry (SOH CAH TOA)

For right-angled triangles, trigonometric ratios relate angles to side lengths.

  • SOH: Sin(angle) = Opposite / Hypotenuse
  • CAH: Cos(angle) = Adjacent / Hypotenuse
  • TOA: Tan(angle) = Opposite / Adjacent

Use these to find unknown sides or angles. Remember to use `sin⁻¹`, `cos⁻¹`, or `tan⁻¹` to find angles.

You must know exact trigonometric values for 0°, 30°, 45°, 60°, and 90°. For example, `sin(30°) = 1/2`, `cos(45°) = √2/2`, `tan(60°) = √3`.

## Non-Right-Angled Triangles

For triangles without a right angle, you use the Sine Rule and Cosine Rule.

Sine Rule

Use the Sine Rule when you have a known side-angle pair and want to find another side or angle.

  • To find a side: `a / sin(A) = b / sin(B) = c / sin(C)`
  • To find an angle: `sin(A) / a = sin(B) / b = sin(C) / c`

Be aware of the ambiguous case when finding an angle using the Sine Rule; there might be two possible angles (θ and 180° - θ).

Cosine Rule

Use the Cosine Rule when you have:

1. Two sides and the included angle (SAS) to find the third side: `a² = b² + c² - 2bc cos(A)`

2. All three sides (SSS) to find an angle: `cos(A) = (b² + c² - a²) / 2bc`

## Area of a Triangle

The area of any triangle can be calculated using the formula: `Area = 1/2 ab sin(C)`, where `a` and `b` are two sides and `C` is the included angle between them.

  • Pythagoras' Theorem: `a² + b² = c²` for right-angled triangles, `c` is the hypotenuse.
  • SOH CAH TOA applies only to right-angled triangles.
  • Sin(angle) = Opposite / Hypotenuse.
  • Cos(angle) = Adjacent / Hypotenuse.
  • Tan(angle) = Opposite / Adjacent.
  • The Sine Rule `a/sin(A) = b/sin(B)` is used for non-right-angled triangles with a known side-angle pair.
  • The Cosine Rule `a² = b² + c² - 2bc cos(A)` finds a side with SAS, or an angle with SSS.
  • Area of any triangle: `1/2 ab sin(C)`, where `C` is the included angle.
  • Memorise exact trig values for 0°, 30°, 45°, 60°, 90°.
What is Pythagoras' Theorem?
`a² + b² = c²`, where `c` is the hypotenuse of a right-angled triangle.
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How do you find an angle in a right-angled triangle using trigonometry?
Use `sin⁻¹`, `cos⁻¹`, or `tan⁻¹` after setting up the SOH CAH TOA ratio.
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What is `sin(45°)`?
`√2/2` (or `1/√2`).
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When would you use the Sine Rule?
In a non-right-angled triangle, when you have a known side-angle pair and want to find another side or angle.
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State the Cosine Rule for finding a side `a`.
`a² = b² + c² - 2bc cos(A)`.
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How do you find the area of a non-right-angled triangle given two sides and the included angle?
`Area = 1/2 ab sin(C)`.
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What is the 'ambiguous case' of the Sine Rule?
When finding an angle, there might be two possible solutions (θ and 180° - θ) if `sin(x)` is positive.
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Circle theorems, vectors and transformations

## Circle Theorems

  • The angle at the centre is twice the angle at the circumference when both are subtended by the same arc.
  • Angles in the same segment are equal.
  • The angle in a semicircle is always 90°.
  • Opposite angles of a cyclic quadrilateral (all four vertices lie on the circumference) sum to 180°.
  • A tangent to a circle is perpendicular to the radius at the point of contact.
  • Two tangents drawn from an external point to a circle are equal in length.
  • The alternate segment theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
  • The perpendicular from the centre to a chord bisects the chord.

## Vectors

  • A vector describes a translation and has both magnitude (size) and direction. It can be written as a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ or using bold letters like a.
  • Adding vectors: Geometrically, place them head-to-tail. Algebraically, add corresponding components: $\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}$.
  • Subtracting vectors: Add the negative vector: a - b = a + (-b).
  • Scalar multiplication: Multiply each component by the scalar: $k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}$.
  • Parallel vectors are scalar multiples of each other (e.g., a = 2b).
  • The magnitude of vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$.
  • Position vectors describe the position of a point relative to the origin O. $\vec{OA}$ is the position vector of A.
  • To prove points are collinear, show that two vectors between them are parallel and share a common point.

## Transformations

  • Rotation: Requires a centre of rotation, an angle (e.g., 90°, 180°), and a direction (clockwise or anti-clockwise).
  • Reflection: Requires a mirror line (e.g., x=2, y=x). Each point is the same perpendicular distance from the line.
  • Translation: Requires a translation vector $\begin{pmatrix} x \\ y \end{pmatrix}$, where x is horizontal movement and y is vertical movement.
  • Enlargement: Requires a centre of enlargement and a scale factor.
  • If scale factor > 1, the object gets larger.
  • If 0 < scale factor < 1, the object gets smaller.
  • If the scale factor is negative, the object is enlarged on the opposite side of the centre and inverted.
  • Combined transformations: Perform transformations one after another. The order often matters.
  • The angle at the centre is twice the angle at the circumference.
  • Opposite angles of a cyclic quadrilateral sum to 180°.
  • A tangent is perpendicular to the radius at the point of contact.
  • Parallel vectors are scalar multiples of each other.
  • The magnitude of vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$.
  • A negative scale factor in enlargement inverts the object through the centre.
  • The angle in a semicircle is always 90 degrees.
  • For rotation, you need a centre, an angle, and a direction.
What is the relationship between the angle at the centre and the angle at the circumference subtended by the same arc?
The angle at the centre is twice the angle at the circumference.
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What is the sum of opposite angles in a cyclic quadrilateral?
180 degrees.
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How do you find the magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$?
$\sqrt{x^2 + y^2}$.
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What does a negative scale factor in an enlargement mean?
The object is enlarged on the opposite side of the centre and inverted.
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What is the alternate segment theorem?
The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
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What three pieces of information are needed to describe a rotation?
Centre of rotation, angle, and direction (clockwise/anti-clockwise).
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How do you prove that three points A, B, C are collinear using vectors?
Show that $\vec{AB}$ is a scalar multiple of $\vec{BC}$ (or $\vec{AC}$) and they share a common point (B).
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Probability, statistics and sampling

## Probability

Probability measures the likelihood of an event occurring, expressed as a fraction, decimal, or percentage between 0 and 1.

  • P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes).
  • Mutually Exclusive Events: Events that cannot happen at the same time. P(A or B) = P(A) + P(B).
  • Independent Events: The outcome of one event does not affect the outcome of another. P(A and B) = P(A) × P(B).
  • Conditional Probability: The probability of an event occurring given that another event has already occurred. Often calculated using tree diagrams (multiplying along branches for 'and', adding branch ends for 'or') or Venn diagrams. P(A|B) = P(A and B) / P(B).
  • Relative Frequency: An estimate of probability based on experimental trials: (Number of times event occurred) / (Total number of trials). As the number of trials increases, relative frequency tends towards theoretical probability.

## Statistics

  • Types of Data: Discrete data can only take specific values (e.g., number of siblings). Continuous data can take any value within a range (e.g., height).
  • Averages & Spread:
  • Mean: Sum of all values / Number of values.
  • Median: The middle value when data is ordered.
  • Mode: The most frequent value.
  • Range: Highest value - Lowest value (measure of spread).
  • From Frequency Tables: For grouped data, the mean is an estimate calculated using the midpoint of each class interval: Σ(midpoint × frequency) / Σfrequency.
  • Cumulative Frequency: Running total of frequencies. Used to draw cumulative frequency graphs to estimate the median, quartiles (lower quartile Q1, upper quartile Q3), and interquartile range (IQR = Q3 - Q1).
  • Histograms: Used for grouped continuous data with unequal class widths. The frequency density is plotted on the y-axis: Frequency Density = Frequency / Class Width. The area of each bar represents the frequency.
  • Scatter Graphs: Show the relationship between two variables. Correlation describes this relationship (positive, negative, or no correlation). A line of best fit can be drawn to make predictions (interpolation within data range, extrapolation outside).

## Sampling

  • A population is the entire group being studied. A sample is a smaller, representative group taken from the population.
  • Random Sampling: Each member of the population has an equal chance of being selected.
  • Stratified Sampling: The population is divided into subgroups (strata), and a proportional number of samples are taken from each stratum. Number in stratum = (Size of stratum / Total population) × Sample size. This ensures the sample is representative of the population's proportions.
  • Bias: Occurs when a sample is not representative of the population, leading to inaccurate conclusions.
  • Probability values always lie between 0 and 1, inclusive.
  • For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
  • For independent events A and B, P(A and B) = P(A) × P(B).
  • Conditional probability P(A|B) is the probability of A given B, calculated as P(A and B) / P(B).
  • Frequency Density = Frequency / Class Width is used for histograms with unequal class widths.
  • The Interquartile Range (IQR) is Q3 - Q1, representing the middle 50% of data spread.
  • To estimate the mean from grouped data, use the midpoint of each class interval.
  • Stratified sampling ensures a sample is proportional to the subgroups within the population.
  • A line of best fit on a scatter graph can be used for interpolation (prediction within data range).
What is the formula for the probability of event A or B if they are mutually exclusive?
P(A or B) = P(A) + P(B)
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How do you calculate frequency density for a histogram?
Frequency Density = Frequency / Class Width
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What is the Interquartile Range (IQR)?
The difference between the upper quartile (Q3) and the lower quartile (Q1). IQR = Q3 - Q1.
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When estimating the mean from a grouped frequency table, what value do you use for each group?
The midpoint of the class interval.
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Describe stratified sampling.
Dividing a population into subgroups (strata) and taking a proportional number of samples from each subgroup.
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What does P(A|B) represent?
The probability of event A occurring given that event B has already occurred (conditional probability).
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What type of data can take any value within a range?
Continuous data.
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How do you calculate the expected number of outcomes for an event?
P(event) × Number of trials.
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