Quadratic equations

26Solve x² – 2x – 15 = 0

Here are two cuboids.

All lengths are measured in centimetres. The volume of cuboid A is 142 cm³ greater than the volume of cuboid B. Work out the value of x.

The point P has coordinates (–4, 5) The point Q has coordinates (6, –6) The point R has coordinates (k, k + 3) Given that angle PRQ is a right angle, find the possible values of k. You must show all your working.

22There are only red counters and yellow counters in a box. of the counters are red. Sophie takes at random two counters from the box. The probability that the two counters are the same colour is Work out the number of yellow counters in the box. You must show all your working.

Solve the simultaneous equations

y + 2 x = 7 Give your solutions correct to 3 significant figures.

21A solid cuboid has a volume of 300 cm³ The cuboid has a total surface area of 370 cm² The length of the cuboid is 20 cm. The width of the cuboid is greater than the height of the cuboid. Work out the height of the cuboid. You must show all your working.

Complete the table of values for y = x² – x

On the grid, draw the graph of y = x² – x for values of x from –2 to 3

Use your graph to find estimates for the solutions of the equation x² – x = 4

Complete the table of values for y = x² – x

On the grid, draw the graph of y = x² – x for values of x from –2 to 3

Use your graph to find estimates for the solutions of the equation x² – x = 4

x – 4, x + 2 and 3x + 1 are three consecutive terms of an arithmetic sequence.

Find the value of x.

y – 4, y + 2 and 3y + 1 are three consecutive terms of a geometric sequence.

Find the possible values of y.

Write x² – 8x + 3 in the form (x – a)² – bwhere a and b are integers.

Hence, write down the coordinates of the turning point on the graph

Solve 7x² + 8x – 5 = 0 Give your solutions correct to 3 significant figures.

Alex has to find the solutions of the quadratic equation 3k² + 10k – 8 = 0 Here is his working and answer. (3 k – 2)(k + 4) = 0 k = 2 or k = –4

What mistake has Alex made?

26Here is the graph of y = x² – 2x – 2

Write down the coordinates of the turning point on the graph of y = x² – 2x – 2

Write down an estimate for one of the roots of x² – 2x – 2 = 0

24Here is the graph of y = x² − 6x + 4

Write down the y intercept of the graph of y = x² − 6x + 4

Write down the coordinates of the turning point of the graph of y = x² − 6x + 4

Use the graph to find estimates for the roots of x² − 6x + 4 = 0

Here is the graph of y = x² − 6x + 4

Write down the y intercept of the graph of y = x² − 6x + 4

Write down the coordinates of the turning point of the graph of y = x² − 6x + 4

Use the graph to find estimates for the roots of x² − 6x + 4 = 0

19Solve 6x² + 5x − 6 = 0

Complete the table of values for y = x² − 3x + 1

On the grid, draw the graph of y = x² − 3x + 1 for values of x from −1 to 4

Using your graph, find estimates for the solutions of the equation x² − 3x + 1 = 0

Complete the table of values for y = x² − 3x + 1

On the grid, draw the graph of y = x² − 3x + 1 for values of x from −1 to 4

Using your graph, find estimates for the solutions of the equation x² − 3x + 1 = 0

Solve x² = 5x + 24

Complete the table of values for y = x² – 2x + 2

On the grid, draw the graph of y = x² – 2x + 2 for values of x from –2 to 4

Use your graph to find estimates of the solutions of the equation x² – 2x + 2 = 4

27Solve x² – 7x – 18 = 0

Complete the table of values for y = x² – 2x + 2

On the grid, draw the graph of y = x² – 2x + 2 for values of x from –2 to 4

Use your graph to find estimates of the solutions of the equation x² – 2x + 2 = 4

16The curve C has equation y = x² + 3x – 3 The line L has equation y – 5x + 4 = 0 Show, algebraically, that C and L have exactly one point in common.

The diagram shows two shaded shapes, A and B. Shape A is formed by removing a sector of a circle with radius (3x – 1) cm from a sector of the circle with radius (5 x – 1) cm. Shape B is a circle of diameter (2 – 2 x) cm.

The area of shape A is equal to the area of shape B. Find the value of x. You must show all your working.

There are only red sweets and yellow sweets in a bag. There are n red sweets in the bag. There are 8 yellow sweets in the bag. Sajid is going to take at random a sweet from the bag and eat it. He says that the probability that the sweet will be red is

Show why the probability cannot be

After Sajid has taken the first sweet from the bag and eaten it, he is going to take at random a second sweet from the bag. Given that the probability that both the sweets he takes will be red is

work out the number of red sweets in the bag. You must show all your working.

Use these graphs to solve the simultaneous equations 5 x − 9 y = −46 y = − 2 x

Use this graph to find estimates for the solutions of the quadratic equation x² − 4x + 2 = 0

Use these graphs to solve the simultaneous equations 5 x − 9 y = −46 y = − 2 x

Use this graph to find estimates for the solutions of the quadratic equation x² − 4x + 2 = 0

The diagram shows a right-angled triangle.

All the measurements are in centimetres.

The area of the triangle is 27.5 cm²

Work out the length of the shortest side of the triangle. You must show all your working.

Here is the graph of y = x² – 3

Use the graph to find estimates for the solutions to the equation x² – 2x – 2 = 0 You must show how you get your solutions.

Given that

find the possible values of x.

There are only green pens and blue pens in a box.

There are three more blue pens than green pens in the box. There are more than 12 pens in the box.

Simon is going to take at random two pens from the box. The probability that Simon will take two pens of the same colour is Work out the number of green pens in the box.

Complete this table of values for y = x² + x – 4

x	–3	–2	–1	0	1	2	3
y		–2			–2

On the grid, draw the graph of y = x² + x – 4 for values of x from –3 to 3

Use the graph to estimate a solution to x² + x – 4 = 0

Expand and simplify (x – 2)(2 x + 3)(x + 1)

Find the value of n.

Solve 5x² – 4x – 3 = 0 Give your solutions correct to 3 significant figures.

f(x) = 4sin x °

Find f(23) Give your answer correct to 3 significant figures.

g(x) = 2 x – 3

Find fg(34) Give your answer correct to 3 significant figures.

h(x) = (x + 4)²

Ivan needs to solve the following equation h(x) = 25

He writes

x + 4 = 5 x = 1

This is not fully correct.

Explain why.

Complete the table of values for y = x² – x – 6

On the grid, draw the graph of y = x² – x – 6 for values of x from –3 to 3

Use your graph to find estimates of the solutions to the equation x² – x – 6 = –2

Complete the table of values for y = x² – x – 6

On the grid, draw the graph of y = x² – x – 6 for values of x from –3 to 3

Use your graph to find estimates of the solutions to the equation x² – x – 6 = –2

ABC is an isosceles right-angled triangle.

The area of the triangle is 162 cm²

Work out the value of x.

Here are two right-angled triangles.

Given that

tan e = tan f

find the value of x.

You must show all your working.

Solve (x – 2)

Give your solutions correct to 3 significant figures.

The diagram shows part of the graph of y = x² – 2x + 3

By drawing a suitable straight line, use your graph to find estimates for the solutions of x² – 3x – 1 = 0

P is the point on the graph of y = x² – 2x + 3 where x = 2

Calculate an estimate for the gradient of the graph at the point P.

Calculate the value of x. Give your answer correct to 3 significant figures.