Quadratic Equations (AQA GCSE Maths: Higher): Exam Questions

Solve    ${\left(x-2\right)}^2=3$

Give your solutions correct to $3$ significant figures.

Solve the equation $3x^2+4x—12=0$

Give your solutions correct to $2$ decimal places.

Solve $3x^2+6x-2=0$

Give your solutions correct to $2$ decimal places.

Solve    $3x^2—x—1=0$

Give your solutions correct to $2$ decimal places.

Solve $x^2—5x+3=0$

Give your solutions correct to $3$ significant figures.

Liz is asked to solve the equation $3x^2+8=83$

Here is her working.

$\begin{array}{rcl}3x^2+8& =& 83\\ 3x^2& =& 75\\ x^2& =& 25\\ x& =& 5\end{array}$

Explain what is wrong with Liz's answer.

The only solution to $x^2+bx+c=0$    is    $x=5$

Work out the values of $b$ and $c$.

$$\begin{gathered} b=.................... \\ c=.................... \end{gathered}$$

Write $x(3x–9)=4$ in the form  $a{x}^2+bx+c=0$ where $a,b$ and $c$are integers.

Solve$x(3x–9)=4$

Give your answers to 2 decimal places.

Solve $x^2-x-12=0$

Without expanding any brackets,

   show how to work out the exact solutions of    $9{(x+3)}^2=4$

Give the solutions.

Solve by factorising.

$x^2+9x+20=0$

$x$= ....................... or $x$ = .......................

Solve this equation algebraically.
Give your solutions correct to 2 decimal places.

3x² + 8x - 5 = 0

x = .................... or x = ....................

Solve.

$x^2-6x+15=3x-5$

$x=$.............. or $x=$..............

Solve.

   $3x^2=75$

$x$ = .................

Solve  $x^2–21x+20=0$

Show your working clearly.

Factorise  $x^2+2x–24$

Hence solve  $x^2+2x–24=0$

Solve     $x^2-6x-8=0$

Write your answer in the form $a\pm \sqrt{b}$ where $a$and $b$are integers.

Alison is using the quadratic formula to solve a quadratic equation.
She substitutes values into the formula and correctly gets.

$x=\frac{-7\pm \sqrt{49-32}}{4}$

Work out the quadratic equation that Alison is solving.
Give your answer in the form $a{x}^2+bx+c=0$, where $a,b$ and $c$ are integers.

Solve   $3x^2-4x-2=0$

Give your solutions correct to $3$ significant figures.

Solve   $3x^2-5x-1=0$

Give your solutions correct to $3$ significant figures.

Solve   $2x^2+3x-7=0$

Give your solutions correct to $2$ decimal places.

The area of square $ABCD$ is $10$ cm². Show that $x^2+6x=1$

Solve $5x^2=10x+4$

Give your answers to 2 decimal places.

$2x^2-6x+5$ can be written in the form $a{(x-b)}^2+c$

where $a,b$ and$c$ are positive numbers.

Work out the values of  $a,b$ and $c$.

$\begin{array}{rcl}a& =& ...................\\ b& =& ...................\\ c& =& ...................\end{array}$

Using your answer to part (a), or otherwise, solve  $2x^2–6x+5=8.5$]

Here are two pieces of work.

For each one, describe the error made and give the complete correct solution.

Error: ........................................................

Correct solution:

Error: .....................................................................

Correct solution:

Solve by factorisation.

$5x^2+7x+2=0$

x = ............................ or x = ............................

Solve by factorisation.

2x² − 19x − 33 = 0

x = ................... or x = .................. 

i) Write $x^2+4x-16$ in the form ${\left(x+a\right)}^2-b$.

[3]

ii) Solve the equation $x^2+4x-16=0$.
Give your answers in surd form as simply as possible.

$x$ = ........... or $x$ = .......... [4]

Solve the equation $3x^2+20x-7=0$

Draw a sketch of the graph with equation $y=3x^2+20x-7$showing all the points of intersection with the coordinate axis

Solve    $x^2=4{\left(x-3\right)}^2$

Solve         $2x^2+9x-7=0$

Give your solutions correct to $3$ significant figures.

Solve      $\frac{2}{y^2}+\frac{9}{y}-7=0$

Give your solutions correct to $3$ significant figures.

Given that

$2x-1:x-4=16x+1:2x-1$

find the possible values of $x$.

The diagram shows a trapezium.

All the measurements are in centimetres.

The area of the trapezium is $351$ cm².

Show that  $2x^2+x-351=0$

Work out the value of $x$.

Given that

$x^2:\left(3x+5\right)=1:2$   

find the possible values of  $x$.

The lengths of the sides are in centimetres.

The area of triangle T₁ is equal to the area of triangle T₂.

Work out the value of $x$, giving your answer in the form  $a+\sqrt{b}$  where $a$ and $b$ are integers.

Solve  $\frac{5}{4x+1}=\frac{2x}{x^2+3}$

Give your solutions to 3 significant figures.
You must show your working.

$y=6x^4+7x^2$ and $x=\sqrt{w+1}$

Find the value of $w$ when $y=10$.
Show your working.

$w$ = ................ 

Given that $M=$ $\frac{{18}^{4n}\times 2^{3\left(n^2-6n\right)}\times 3^{2\left(1-4n\right)}}{{12}^2}$ 

find the values of $n$ for which $M=2$

The diagram shows a trapezium.

All measurements shown on the diagram are in centimetres. The area of the trapezium is $133{\mathrm{cm}}^2$

Show that $8x^2–6x–275=0$

Find the value of $x.$

Show your working clearly.

Express $7–4x–x^2$ in the form $p–{(x+q)}^2$ where $p$ and $q$ are constants.

Use your answer to part (a) to solve the equation $7–4(y+3)–{(y+3)}^2=0$

Give your solutions in the form $e\pm \sqrt{f}$ where $e$ and$f$ are integers.

$6x^2=7xy+20y^2$  where  $x>0$ and  $y>0$

Find the ratio  $x:y$