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

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 $x^2+6x=1$

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 $5x^2=10x+4$

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  $2x^2–6x+5=8.5$

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.

Mira normally cycles a $10\text{ km}$ route at a steady speed of $v\text{ km/h}$.

One day, she cycles the same route at a steady speed that is $2\text{ km/h}$ faster than usual, and the journey takes her $15$ minutes less than usual.

Find the value of $v$.

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.

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

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

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

Find the ratio  $x:y$