Quadratic Formula Questions | Worksheets and Revision | MME

Quadratic Formula Questions, Worksheets and Revision

Level 6 Level 7
GCSE Maths Revision Cards

What you need to know

The quadratic formula:

The quadratic formula is a formula that you can substitute values into in order to get the solutions to any quadratic equation.

x=\dfrac{-\textcolor{red}{b}\pm\sqrt{\textcolor{red}{b}^2-4\textcolor{blue}{a}\textcolor{Orange}{c}}}{2\textcolor{blue}{a}}

The \textcolor{blue}{a}, \textcolor{red}{b} and \textcolor{Orange}{c} values correspond to values in a quadratic equation, shown below

\textcolor{blue}{a}x^2+\textcolor{red}{b}x+\textcolor{Orange}{c}=0

Take note:

Notice that there is a “plus or minus” symbol in there {\textcolor{blue}{(\pm)}}.

This is because a quadratic has up to two real solutions – putting a plus sign there will give you one solution and putting a minus sign there will give you the other. In other words, the two solutions are

x=\dfrac{-b\textcolor{blue}{+}\sqrt{b^2-4ac}}{2a} and x=\dfrac{-b\textcolor{blue}{-}\sqrt{b^2-4ac}}{2a}

You are not given this formula in an exam so you do have to memorise it.

Example 1:

Use the quadratic formula the following quadratic equation:

x^2+2x-35=0

[2 marks]

Firstly, we have to identify what a,b, and c are:

a=1.

b=2.

c=-35.

Next we need to substitute these into the formula:

x=\dfrac{-2\pm\sqrt{2^2-4\times1\times(-35)}}{2}

Simplifying this we get

x=\dfrac{-2+\sqrt{144}}{2} and x=\dfrac{-2-\sqrt{144}}{2}

We know \sqrt{144}=12, so the two final solutions are

x_1=\dfrac{-2+12}{2}=5 and x_2=\dfrac{-2-12}{2}=-7

Example 2:

Use the quadratic formula to solve the following quadratic equation:

2x^2-6x+3=0

Give your answer to 2 decimal places.

[2 marks]

a=2

b=-6

c=3

Putting these into the formula, we get

x=\dfrac{-(-6)\pm\sqrt{(-6)^2-4\times2\times3}}{2\times2}

so, the solutions are

x=\dfrac{6+\sqrt{12}}{4} and x=\dfrac{6-\sqrt{12}}{4}

12 is not a square number, which is how we know this won’t give us a nice answer.

The question asks for 2dp, so putting these into the calculator, we get

x_1=2.366...=2.37 2dp and x_2=0.6339...=0.63 2dp

Note: You can put the first quadratic formula straight into the calculator without any simplifying and use the + and – to get your two answers. 

Example Questions

Firstly, the quadratic formula is

 

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

 

Then, we can identify that here, a=1, b=11, and c=16. Putting these values into the formula, we get

 

x=\dfrac{-11\pm\sqrt{11^2-4\times1\times16}}{2}

 

The part inside the square root is

 

11^2-4\times1\times16=121-64=57

 

So, the solutions become

 

x=\dfrac{-11\pm\sqrt{57}}{2}

 

Putting these into a calculator (one with +, one with -), we get the final solutions

 

x=-1.7250...=1.73\text{ (3sf), and }\,x=-9.2749...=-9.27\text{ (3sf)}

Firstly, the quadratic formula is

 

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

 

Then, we can identify that here, a=1, b=-2, and c=-44. Putting these values into the formula, we get

 

x=\dfrac{-(-2)\pm\sqrt{(-2)^2-4\times1\times(-44)}}{2}

 

The part inside the square root is

 

(-2)^2-4\times1\times(-44)=4+176=180

 

So, the solutions become

 

x=\dfrac{2\pm\sqrt{180}}{2}

 

Putting these into a calculator (one with +, one with -), we get the final solutions

 

x=7.7082...=7.71\text{ (3sf), and }\,x=-5.7082...=-5.71\text{ (3sf)}

Firstly, the quadratic formula is

 

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

 

Then, we can identify that here, a=4, b=7, and c=-1. Putting these values into the formula, we get

 

x=\dfrac{-7\pm\sqrt{7^2-4\times4\times(-1)}}{2\times4}

 

The part inside the square root is

 

7^2-4\times4\times(-1)=49+16=65

 

So, the solutions become

 

x=\dfrac{-7\pm\sqrt{65}}{8}

 

Putting these into a calculator (one with +, one with -), we get the final solutions

 

x=0.1327...=0.13\text{ (2dp), and }\,x=-1.8827...=-1.88\text{ (2dp)}

Firstly, the quadratic formula is

 

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

 

Then, we can identify that here, a=1, b=8, and c=13. Putting these values into the formula, we get

 

x=\dfrac{-8\pm\sqrt{8^2-4\times1\times13}}{2\times1}

 

The part inside the square root is

 

8^2-4\times8\times13=64-52=12

 

So, the solutions become

 

x=\dfrac{-8\pm\sqrt{12}}{2}

 

Putting these into a calculator (one with +, one with -), we get the final solutions

 

x=-2.2679...=-2.27\text{ (2dp), and }\,x=-5.73205...=-5.73\text{ (2dp)}

Firstly, the quadratic formula is

 

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

 

Then, we can identify that here, a=25, b=-30, and c=7. Putting these values into the formula, we get

 

x=\dfrac{-(-30)\pm\sqrt{(-30)^2-4\times25\times7}}{2\times25}

 

The part inside the square root is

 

30^2-4\times25\times7=900-700=200

 

So, the solutions become

 

x=\dfrac{30\pm\sqrt{200}}{50}

 

Putting these into a calculator (one with +, one with -), we get the final solutions,

 

x=\dfrac{3\pm\sqrt{2}}{5}

Worksheets and Exam Questions

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(NEW) The Quadratic Formula Exam Style Questions - MME

Level 6-7
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Quadratic Formula (1) - Drill Questions

Level 6-7
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Quadratic Formula (2) - Drill Questions

Level 6-7

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