Quadratic equations

The table below shows some of the values of y = 2x² − 6 for values of x from –3 to 2.

Complete the table above.

On the graph paper opposite, draw the graph of y = 2x² − 6 for values of x

from –3 to 2.

Factorise 2x² + 3x − 20.

Hence solve the equation 2x² + 3x − 20 = 0.

Use the cosine rule to show that 3x² +10x − 35 = 0.

Solve the equation

You must use the quadratic formula and show all your working.

Give your answers correct to 2 decimal places.

Evaluate the length of PQ.

You must justify any decision that you make.

Use your solutions from part (b) to find the dimensions of the rectangle.

You must justify any decisions that you make.

Use the quadratic formula to solve the equation 11x² = 23x +19. Give your answer correct to 2 decimal places.

You must show all your working.

Factorise x² + 3x − 40, and hence solve x² + 3x − 40 = 0.

Factorise x² + 3x − 40, and hence solve x² + 3x − 40 = 0.

Solve the equation 7 x (x + 3) + 1 = x (x + 2).

You must use the quadratic formula and show all your working.

Give your answers correct to 2 decimal places.

Factorise x² −8x − 20, and hence solve x² −8x − 20 = 0.

Factorise x² −8x − 20, and hence solve x² −8x − 20 = 0.

Use the quadratic formula to solve the equation + 5 = 7x.

Give your answers as decimal numbers.

You must show all your working.

Show that 16x² 8x 3 0.

Solve the equation 16x² 8x 3 0.

You must use an algebraic method.

Find the length of BC.

You must justify any decision that you make.

Factorise 8x² 18.

Hence solve 8x² 18 0.

Hence, sketch the graph of y 8x² 18 on the axes below.

Mark clearly the coordinates of any point where this graph crosses an axis.

Space for working:

Use the quadratic formula to solve the equation 1.

Give your answers correct to 2 decimal places.

You must show all your working.

Solve the following equation.

Do not use a trial and improvement method.

The table below shows some of the values of y x² x 4 for values of x from –3 to 3.

x	–3	–2	–1	0	1	2	3
y x2 x 4	2	–2		– 4		2	8

Complete the table by finding the values of y for x = –1 and for x = 1.

On the graph paper opposite, draw the graph of y x² x 4 for values of x from

–3 to 3.

The table below shows some of the values of y x² x 4 for values of x from –3 to 3.

x	–3	–2	–1	0	1	2	3
y x2 x 4	2	–2		– 4		2	8

Complete the table by finding the values of y for x = –1 and for x = 1.

Marian is competing in a race.

The race is (6x + 5) miles long. She completes the race in x hours. Her average speed during the race is (2x + 3) miles per hour.

Calculate how long Marian takes to complete the race.

You must use an algebraic method (not trial and improvement).

Shape A is a square and Shape B is a rectangle, as shown below.

The area of square A is equal to the area of rectangle B.

Form an equation in x and hence calculate the value of x. Give your answer correct to 1 decimal place.

You must show all your working.

Factorise x² − 4x − 12, and hence solve x² − 4x − 12 = 0.

Solve the equation

You must use an algebraic method and show all your working.

Factorise x² − 4x − 12, and hence solve x² − 4x − 12 = 0.

Show that

Solve the equation

You must use an algebraic method and show all your working.

Give your answers correct to 2 decimal places.

Evaluate the length of AC.

You must justify any decision that you make.

Factorise x² – 7x + 12, and hence solve x² – 7x + 12 = 0.

Expand and simplify (5x – 2)².

The table below shows some of the values of y = x² − 4x − 3 for values of x from –2 to 5.

Complete the table by finding the value of y for x = −2 and the value of y for x = 2.

On the graph paper opposite, draw the graph of y = x² − 4x − 3 for values of x from

–2 to 5.

Draw the line y = 1 on the graph paper.

Write down the values of x where the line y = 1 cuts the curve y = x² − 4x − 3.

Values of x are_........ and_........

Factorise x² – 7x + 12, and hence solve x² – 7x + 12 = 0.

Expand and simplify (5x – 2)².

Use the quadratic formula to solve (5 x + 3)(5 x – 3) = 19 x.

Give your answers correct to 2 decimal places.

You must show all your working.

Factorise x² + 4x – 21. Hence, solve x² + 4x – 21 = 0.

Factorise x² + 4x – 21. Hence, solve x² + 4x – 21 = 0.

The table below shows some of the values of y = x² − 2x − 4 for values of x from –3 to 4.

Complete the table by finding the value of y when x = 1.

On the graph paper opposite, draw the graph of y = x² − 2x − 4 for values of x from

–3 to 4.

Draw the line y + x = 4 on the graph paper.

Show that 2x² + 3x – 50 = 0.

Solve the equation 2x² + 3x – 50 = 0 to calculate the lengths AB and DC.

Give your answers correct to 1 decimal place.

Length of AB =_{……........ ……..} cm Length of DC =_{……........ ……..} cm

A rhombus has an area of 36∙8 cm². The rhombus is enlarged by a scale factor of 7.

Calculate the area of the enlarged rhombus.

Write the following expression as a single fraction.

Give your answer in its simplest form.

Solve the following equation.

Use the quadratic formula to solve (3x – 2)² = (x + 1)(x + 2).

Give your answers correct to 2 decimal places.

You must show all your working.

The table below shows some of the values of y = x² + 4x − 1 for values of x from –5 to 2.

Complete the table by finding the value of y for x = –2 and for x = 2.

On the graph paper opposite, draw the graph of y = x² + 4x − 1 for values of x from

–5 to 2.

The table below shows some of the values of y = x² + 4x − 1 for values of x from –5 to 2.

Complete the table by finding the value of y for x = –2 and for x = 2.

On the graph paper opposite, draw the graph of y = x² + 4x − 1 for values of x from

–5 to 2.

Draw the line y = 2 on the graph paper.

Write down the values of x where the line y = 2 cuts the curve y = x² + 4x − 1.

Values of x are_........ and_........

Give your answers correct to 2 decimal places.

You must show all your working.

A ball is thrown upwards from a height of 1 m above the ground.

After t seconds, its height above the ground is h metres, where h is given by

Show that the time taken to reach a height of 4 metres satisfies the equation

5t² – 8t + 3 = 0.

Solve the equation

5t² – 8t + 3 = 0.

Interpret your answer to part (b) in the context of the question.

Five quadratic equations are listed below.

Draw a line connecting each equation to its solution.

One has been completed for you.

Solve the equation (2 x + 5)(3 x – 11) = 7.

Give your answers correct to 2 decimal places.

Factorise x² – 7x – 18, and hence solve x² – 7x – 18 = 0.

Factorise x² – 7x – 18, and hence solve x² – 7x – 18 = 0.

Two different squares are constructed. The side length of the smaller square is xcm. The side length of the larger square is 3 cm longer than the side length of the smaller square. The combined area of the two squares is 22∙5 cm².

Show that 4x² + 12x − 27 = 0.

Show that (10w + 3)(w – 1) – (2 – 3w)² w² + 5w – 7.

Use the quadratic formula to solve the equation w² + 5w – 7 = 0.

Give your answers correct to 2 decimal places.

The table below shows some of the values of y = x² – 5x + 2, for values of x from –1 to 5.

Complete the table above.

On the graph paper below, draw the graph of y = x² – 5x + 2 for values of x from

Draw the line y = –3 on the graph paper.

Write down the values of x where the line y = –3 cuts the curve y = x² – 5x + 2.

Give your answers correct to 1 decimal place.

The diagram shows two rectangles.

The combined area of both rectangles is 50 cm². By considering the areas of the two rectangles, show that 2x² – 5x – 25 = 0 and hence find the

value of x.