- A-Level Maths

# Quadratic & Simultaneous Equations and Inequalities

The standard format of a quadratic expression is **ax² + bx + c**

There are three ways of solving quadratic equations:

### Factorising

Factorise a quadratic in the form *ax² + bx + c = 0*, and set each bracket to equal 0 to find the values of x (the roots).

### The Quadratic Equation

The** (b² - 4ac)** inside the square root is known as the **discriminant**, and is used to show how many roots a quadratic has:

**b² - 4ac > 0:**The quadratic has**two distinct roots****b² - 4ac = 0:**The quadratic has**one repeated root****b² - 4ac < 0:**The quadratic has**no real roots**

### Completing the Square

More commonly, quadratics are in their standard form. In this case, this version is used:

## Simultaneous Equations

### Linear Simultaneous Equations

There are two ways of solving linear simultaneous equations: **elimination** and **substitution**.

For example, solve the following simultaneous equations:

**x + 3y = 11**

**4x - 5y = 10**

**Elimination**

Multiply the first equation by 4

4x + 12y = 44

4x - 5y = 10

Subtract

17y = 34

**y = 2**Substitute this into equation 1

x + 6 = 11

**x = 5**

**Substitution**

Rearrange the first equation to make x the subject

x = 11 - 3y

Substitute this into the second equation and solve

4(11 - 3y) - 5y = 10

44 - 12y - 5y = 10

44 - 17y = 10

-17y = -34

**y = 2**Substitute this into the rearranged equation 1

x = 11 - 3(2)

**x = 5**

### Quadratic Simultaneous Equations

Two simultaneous equations, one linear and one quadratic, can have up to two pairs of solutions. **Don't get confused between the tow pairs!**

You always use the **substitution method **above to solve quadratic simultaneous equations - rearrange the linear equation and sub into the quadratic:

Solve the simultaneous equations, **x + 2y = 3**, and **x² +3xy = 10**

Rearrange linear equation to make x the subject

x = 3 - 2y

Substitute into quadratic equation & solve

(3 - 2y)² + 3y(3 - 2y) = 10

9 - 12y + 4y² + 9y - 6y² = 10

9 - 3y - 2y² = 10

2y² + 3y +1 = 0

(2y +1)(y +1) = 0

**y = -1/2, -1**Find the corresponding x values

x = 3 - 2(-1/2)

**x = 4**x = 3 - 2(-1)

**x = 5**Get the pairs together correctly:

**x = 4, y = -1/2 and x = 5, y = -1**

### Graphing Simultaneous Equations

The solutions to a pair of simultaneous equations represents the intersections between their graphs. For a linear and quadratic pair of simultaneous equations, you can use the discriminant of the substituted equation (the linear equation substituted into the quadratic equation) to show whether or not there are any solutions, and if so, how many.

## Inequalities

There is certain notation for inequalities on a number line:

Linear inequalities are rearranged to make the variable the subject.

**To solve a quadratic inequality:**

rearrange so that to the right of the inequality sign is 0

solve the remaining quadratic on the left

Sketch this equation roughly to see where the roots are and if it is positive or negative

Identify the correct section.

### Regions on Graphs

It is possible to show regions closed off by one or multiple lines on a graph. Again, there is certain notation to be aware of:

**Dashed lines**do**not**include the curve**Solid lines**do include the curveShaded areas represent the defined region