Quadratic equations

Solve.

8 x - 11 = 6 x + 3

Solve by factorising.

x² + 15x - 16 = 0

Solve.

8 x - 11 = 6 x + 3

Solve by factorising.

x² + 15x - 16 = 0

Solve.

You must show your working.

In this question, all lengths are in centimetres.

The area of the rectangle is 70 cm².

Show that x² + 5x - 84 = 0.

Solve by factorising.

(b)(i) x =........ or x =........

Find the length of the longer side of the rectangle.

Solve by factorisation.

Write x² + 8x + 11 in the form (x + a)² - b.

Write down the coordinates of the turning point of the graph y = (x - 3)² + 8.

Describe the single transformation which maps the graph of y = x² onto the graph of y = (x - 3)² + 8.

The sketch shows the coordinate axes and the point (0, 10).

The distance from the point (0, 10) to a point (12, t) is 12.5 units.

Work out the two possible values of t. You must show your working and you may use the sketch to help.

Multiply out.

5 a (2 - a)

Solve.

5 x + 9 > 12

Solve by factorising.

x² + 7x + 12 = 0

Solve this quadratic equation by factorisation.

2x² - 6x - 24 = 5x - 3

Sasha and Taylor are asked to find how many solutions the equation 5(x + 2)² = 45 has.

Here is Sasha’s answer.
Here is Taylor’s answer.

Decide who is correct, Sasha or Taylor, and give the reason for your decision.

........ is correct because........

Solve this equation algebraically. Give your answers correct to 2 decimal places. You must show your working.

x² - 5x + 3 = 0

The diagram shows a prism of length 10 cm.

The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the height, h cm. The volume of the prism is 240 cm³.

A student is explaining how they worked out the value of b.

They say

b is 6 because that means h is 4 and 6 × 4 × 10 = 240.

Describe the student’s error and find the correct value of b.

Factorise x² + 10x + 24.

Write down the solutions to x² + 10x + 24 = 0.

Here are two pieces of work.

Each shows a question and the first line of an incorrect solution.

For each part, describe the error made in the first line of the solution. You do not need to complete the solution.

The diagram shows a prism of length 10 cm.

The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the height, h cm. The volume of the prism is 240 cm³.

A student is explaining how they worked out the value of b.

They say

b is 6 because that means h is 4 and 6 × 4 × 10 = 240.

Describe the student’s error and find the correct value of b.

Factorise.

9x² - 4

Solve by factorisation.

3x² - 2x - 8 = 0

Solve.

Solve.

x² - 4x - 165 = 0

You must show your working.

By factorising, find the roots of y = x² + 18x + 77.

(a) x =........ and x =........

Write y = x² + 18x + 77 in the form y = (x + a)² - b.

Write down the coordinates of the turning point of the graph of y = x² + 18x + 77.

Darcy says that the statement in the box is an equation. Ellis says that the statement in the box is an identity. One of them is correct.

Explain which one of Darcy or Ellis is correct.

........ is correct because........

Solve by factorising.

x² + 4x - 12 = 0

The graph of y = x² + 6x - 2 is shown below. The roots of the equation x² + 6x - 2 = 0 are at p and q.

Calculate y when x = 1.

Without solving the equation, explain why q must lie between 0 and 1.

Explain why using a method of iteration is not the most appropriate way of finding a solution to this equation.

Charlie is making some wooden frames. Charlie has a strip of wood 1.6 m long and 2 cm wide.

Not to scale

Each frame will be made from four pieces of wood cut from the strip to form a rectangle, as shown below.

The width of each frame is x cm. The length of each frame is 2 x cm. The area enclosed by each frame must be 96 cm².

Work out the maximum number of frames Charlie can make from the 1.6 m length of wood. You must show your working.

Write x² - 8x + 9 in the form x - a² - b.

Use your answer from part (a) to solve.

x² - 8x + 9 = 0

Give your answers in exact form. You must show your working.

In this question, all measurements are in centimetres.

The square and the rectangle have the same area.

Show that x² - 8x - 20 = 0.

Solve x² - 8x - 20 = 0.

Explain why one of the answers in part (b) is not possible in the context of the question.

Write down the following.

The area of the square.

The length of the rectangle.

The following kinematics formulas may be used in this question.

v = u + at

v² = u² + 2as

The initial velocity of a particle is 20 m/s. The acceleration of the particle is -8 m/s². After t seconds, the particle has travelled 25 m.

Show that 4t² - 20t + 25 = 0.

Solve 4t² - 20t + 25 = 0.

Show that the particle is stationary when it has travelled 25 m.

Solve.

Factorise.

5a² - 10a

Solve by factorising.

x² + 15x + 56 = 0

Here are two pieces of work.

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

Correct solution:

Correct solution:

Solve by factorising.

Solve by factorising.

x² + 3x - 10 = 0

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

Solve by factorisation.

Multiply out.

Give your answer in its simplest form.

32x + d + cx + 5 = 10x + 17

Work out the value of c and the value of d.

Solve by factorising.

Solve by factorisation.

Write x² + 4x - 16 in the form (x + a)² - b.

Solve the equation x² + 4x - 16 = 0. Give your answers in surd form as simply as possible.

(ii) x =........ or x =........

Sketch the graph of y = x² + 4x - 16, showing clearly the coordinates of any turning points.

In the diagram, the square and the trapezium share a common side of length x cm.

The area of the square is equal to the area of the trapezium.

Work out the value of x.

Solve.

4 x = 56

8 x - 6 = 46

Solve by factorising.

Solve by factorisation.

3x² + 11x – 20 = 0

Solve this equation, giving your answers correct to 1 decimal place.

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

3x² + 5x - 1 = 0

Simplify.

7 t - 6 u + 5 t - 4 u

Factorise.

5 v + 20 w

Solve by factorising.

Solve by factorisation.

Solve this equation. Give each value correct to 2 decimal places.