Quadratic equations

The area of the rectangle is 120 cm²

Work out the value of x.

x =

The area of the rectangle is 120 cm²

Work out the value of x.

There are n counters in a box. 6 of the counters are black. Two counters are chosen at random without replacement. The probability that both counters are black is

Use an algebraic method to work out the value of n.

n =

Solve 3x² + 5x – 9 = 0 Give your solutions as decimals.

Solve (x + 2)(x – 5) = 6 x

9k + 7 and 2k² + 3 are consecutive integers. 9 k + 7 is the smaller integer.

Work out the value of the next consecutive integer.

x is a square number.

Work out one set of possible values for d, e and f.

e =

f =

f(x) = x – 3 g(x) = 4 x – 7

Work out the value of fg(6)

Solve (f(x))² = g(x)

Here is the graph of y = 0.5x² – 6x + 12

Use the graph to estimate the solutions of 0.5x² – 6x + 12 = 0

Here is the graph of y = 0.5x² – 6x + 12

Use the graph to estimate the solutions of 0.5x² – 6x + 12 = 0

Solve 2 x (x + 10) = 5 x – 18

Factorise x² + 8x + 15

Write down the two solutions of (y + 2)(y – 4) = 0

f(x) = x² + 6x g(x) = 2 x + 4

Show that fg(x) = 4x² + 28x + 40

Solve fg(x) = – 5

The only solution to x² + bx + c = 0 is x = –15 Work out the values of b and c.

Using the quadratic formula, or otherwise, solve 3x² + x – 5 = 0

In a class there are n boys a total of 25 students. Two of the students are chosen at random. The probability that both students are boys is Work out the value of n.

Solve x² + 7x – 11 = 0 Give your solutions as decimals.

Here is a sketch of the curve y = x² – 4x – 5

Write down the two roots of x² – 4x – 5 = 0

Work out the coordinates of T, the turning point of the curve.

Here is the plan of the floor of an L-shaped room. All lengths are in metres.

The area of the floor is 75 m²

By factorising x² + x – 90 work out the value of x. You must show your working

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

Kate has the following question for homework.

Show that Kate can form the equation x 2 + 5 x – 1400 = 0

Kate correctly factorises the equation to get (x + 40)(x – 35) = 0 Her answer to the homework question is x = – 40 or x = 35 Is her answer correct? Tick a box.

Give a reason for your answer.

Work out the two roots of (7 x + 1)(2 x – 3) = 0 Circle both roots.

Toby is forming and solving equations.

The product of half of a number and three more than the number is the same as the square of the number Toby uses y to represent the number. Write an equation that Toby could form.

Toby forms another equation.

He wants to work out the values of x. Here is his working.

What error has he made in his working?

Circle the two roots of (x – 5)(x + 3) = 0

–5 –3 3 5

Write x(3x – 9) = 4 in the form ax² + bx + c = 0 where a, b and c are integers.

Solve x (3 x – 9) = 4 Give your answers to 2 decimal places.

Solve x² – x – 12 = 0

Solve x² – x – 12 = 0

Jo wants to work out the solutions of x² + 3x – 5 = 0 She says, ‘‘The solutions cannot be worked out because x² + 3x – 5 does not factorise to (x + a)(x + b) where a and b are integers.’’ Is Jo correct? Tick a box.

Give a reason for your answer.

Without expanding any brackets, show how to work out the exact solutions of 9(x + 3)² = 4 Give the solutions.

Circle the two roots of (2 x + 3)(5 x – 2) = 0

Meera is using a graphical method to solve 2x² – 3x = 0 She draws the graph of y = 2x² and a straight line graph on the same grid.

Complete her method to solve 2x² – 3x = 0

Levi is solving 2x² + 5x = 0 He uses this method. 2x² + 5x = 0 subtract 5x from both sides 2x² = –5x divide both sides by x 2 x = –5 divide both sides by 2 x = –2.5 Evaluate his method and his answer.

Give your solutions to 2 decimal places. You must show your working.

Solve 5x² = 10x + 4 Give your answers to 2 decimal places.