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Factorising Factorising quadratics Expanding brackets Simplifying algebraic expressionsThis topic is relevant for:

Here we will learn about the **quadratic formula **and how we can use it to solve **quadratic equations**.

There are also solving quadratic equations worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

If a quadratic algebraic equation will not factorise, we can use the **quadratic formula** to solve quadratic equations.

\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

By using the general form of a quadratic equation

\[a x^{2}+b x+c=0\]

we can substitute the values of

We need to memorise the quadratic formula.

Quadratic equations normally have two solutions, so we need to use the formula twice, once with a + and once with a −.

\[\begin{aligned}
x=\frac{-b+\sqrt{b^2-4ac}}{2a}\\
\\
x=\frac{-b-\sqrt{b^2-4ac}}{2a}
\end{aligned}\]

The solutions to quadratic equations could involve fractions, decimals or integers.

Get your free quadratic formula worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free quadratic formula worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONIn order to solve a quadratic equation by using the quadratic formula, it is a good idea to simplify it and make sure it is in the form of the general quadratic equation.

\[ax^{2}+bx+c=0\]

We must ensure the quadratic equation is equal to 0, rearranging it if necessary.

Step by step guide:

- Identify the value of
a ,b andc . - Substitute these values into the quadratic formula.
- Use a calculator to solve the equation with a +, and then with a −.

To solve

\[x^{2}-8x+15=0\]

**Identify the a, b and c**.

\[a=1,\qquad b=-8,\qquad c=15\]

2**Substitute these values into the quadratic formula**.

\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

\[x=\frac{-(-8)\pm\sqrt{(-8)^2-4(1)(15)}}{2(1)}\]

Using brackets will help to make the calculation clear.

3**Use a calculator to solve the equation with a +, and then with a −**.

\[\begin{aligned}
x&=\frac{-(-8)+\sqrt{(-8)^2-4(1)(15)}}{2(1)}\\
x&=5\\
\\
\\
x&=\frac{-(-8)-\sqrt{(-8)^2-4(1)(15)}}{2(1)}\\
x&=3
\end{aligned}\]

When we plot the graph of the quadratic equation we get a special ‘U’ shaped curve called a parabola.

By using graphing techniques such as this we can see that the roots of the quadratic equation are where the quadratic graph **crosses the x-axis**.

We can check that our solution is correct by substituting it back into the quadratic function.

To solve

\[2x^{2}+5x+3=0\]

**Identify the a, b and c**.

\[a=2,\qquad b=5,\qquad c=3\]

**Substitute these values into the quadratic formula**.

\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

\[x=\frac{-(5)\pm\sqrt{(5)^2-4(2)(3)}}{2(2)}\]

Using brackets will help to make the calculation clear.

**Use a calculator to solve the equation with a +, and then with a −**.

\[\begin{aligned}
x&=\frac{-(5)+\sqrt{(5)^2-4(2)(3)}}{2(2)}\\
x&=-1\\
\\
\\
x&=\frac{-(5)-\sqrt{(5)^2-4(2)(3)}}{2(2)}\\
x&=-1.5
\end{aligned}\]

We can see the roots of the quadratic equation are where the quadratic graph crosses the x-axis.

We can check that our solution is correct by substituting it into the original equation.

**When to use the quadratic formula**

We can use the quadratic formula to solve quadratic equations regardless of whether the factorising (sometimes called factoring) method will work.

**Not including the sign in front of**a ,b andc

When writing the coefficients of the variables for the values of

\[x^{2}-8x+15=0\]

So here

**Typing into calculator**

Mistakes can be made when typing into a calculator.

E.g. for

\[2x^{2}+5x+3=0\]

\[a=2,\qquad b=5,\qquad c=3\]

Use the fraction button and brackets to carefully type into the calculator.

Use the arrow buttons to change the + on the top left of the calculation to a − to work out the second solution.

**Sometimes there are no real roots to a quadratic equation**

E.g.

\[x^{2}-4 x+5=0\]

We can see that there are no values of x that give a y value of 0.

The graph does not cross the x-axis.

There is an easy way to find out how many real roots a quadratic equation has by using the expression under the square root in the quadratic formula.

\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

\[b^{2}-4ac\]

This is called the **discriminant**.

- If

\[b^{2}-4 a c>0\]

this means there are two solutions (these are real numbers).

- If

\[b^{2}-4 a c=0\]

this means there is one solution (which is a real number).

- If

\[b^{2}-4 a c<0\]

this means there are no real solutions, here the solutions will be imaginary numbers.

1. Solve

{x}^2+4x+3=0

Show answer

x=-3 , x=-1

2. Solve

{x}^2+3x-18=0

Show answer

x=-6 , x=3

3. Solve

2{x}^2-3x-2=0

Show answer

x=-0.5 , x=2

4. Solve

3{x}^2-4x-5=0

Round your answers to 3 significant figures

Show answer

x=2.12 , x=-0.786

1. Solve:

x^{2}+2x-4=0

Give your answer to 2.d.p

**(3 marks)**

Show answer

Substitution into the quadratic formula

**(1)**

x=1.24 (2.d.p)

**(1)**

x=-3.24 (2.d.p)

**(1)**

2. Solve:

2x^{2}-3x+1=0

**(3 marks)**

Show answer

Substitution into the quadratic formula or an attempt to factorise

**(1)**

x=1

**(1)**

x=\frac{1}{2} \quad

**(1)**

3. Solve:

3 x^{2}+6 x-2=0

Give your answer to 2.d.p

**(3 marks)**

Show answer

Substitution into the quadratic formula

**(1)**

x=0.29 (2.d.p)

**(1)**

x=-2.29 (2.d.p)

**(1)**

- Did you know that Al-Khwarizmi (Abu Ja'far Muhammad ibn Musa al-Khwarizmi
**)**was one of the first people in history to write about algebra? He lived in Baghdad around 780 to 850 AD and his book "Hisab**Al-jabr**w’al-muqabala" is where we get the word ‘**algebra’**(meaning 'restoration of broken parts').

- Did you know the ancient Babylonians could solve quadratic equations using a method equivalent to the
**quadratic formula**, despite not using algebraic notation!

The history of mathematics is amazing!

- Solve quadratic equations algebraically by using the quadratic formula (H)
- Solve quadratic equations by finding approximate solutions using a graph

- Quadratic equations (factorising)
- Completing the square
- Simultaneous equations

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