Quadratic equations

Show clear algebraic working.

Factorise y² – 11y + 30

Hence solve y² – 11y + 30 = 0

Expand x(x 3)

Simplify a⁶ a¹⁰

Simplify c³⁰ c¹²

Factorise y² 10y 21

Hence, solve y² 10y 21 0

6

Show clear algebraic working.

Factorise y² – 11y + 30

Hence solve y² – 11y + 30 = 0

Simplify a⁶ a¹⁰

Simplify c³⁰ c¹²

Factorise y² 10y 21

Hence, solve y² 10y 21 0

19Solve the inequality 4x² 4x 15 0 Show clear algebraic working.

The diagram shows a cuboid.

The cuboid measures 3x cm by 2x cm by y cm

The volume of the cuboid is 1014cm³ The total surface area of the cuboid is A cm²

You must show all the stages of your working.

Factorise x² 5x 24

Hence, solve x² 5x 24 0

Solve the inequality 3y 5 7y 10

Show clear algebraic working.

Simplify g⁹ g²

Expand 5k²(k³ + 4)

Factorise x² 2x 63

Hence, solve x² 2x 63 = 0

Solve the inequality 7 2y 3y 12

Simplify g⁹ g²

Expand 5k²(k³ + 4)

Factorise x² 2x 63

Hence, solve x² 2x 63 = 0

Solve the inequality 7 2y 3y 12

There are 25 counters in a bag such that

6 counters are blue x counters are orange, where x > 9 the rest of the counters are pink

Maalam takes at random two of the counters from the bag. The probability that Maalam takes one orange counter and one pink counter is Calculate the probability that Maalam takes 2 pink counters from the bag. Show clear algebraic working.

24A polygon has n sides, where n 5

The interior angles of the polygon form an arithmetic sequence.

The smallest angle of the polygon is 84° The common difference of the sequence is 4°

Work out the sum of the interior angles of the polygon. Show clear algebraic working.

The diagram shows a hexagon ABCDEF

AB = 25 cm BC = (x + 2) cm CD = 8 cm EF = 7 cm AF = (x + 6) cm

The area of hexagon ABCDEF is 258 cm²

Work out the value of x

Simplify (2c⁴d⁷)³

Find the value of 5y⁰ where y 0

Factorise fully 16a²b³ + 20a³b

Factorise x² + 9x – 22

Hence solve x² + 9x – 22 = 0

Simplify (2c⁴d⁷)³

Find the value of 5y⁰ where y 0

Factorise fully 16a²b³ + 20a³b

Factorise x² + 9x – 22

Hence solve x² + 9x – 22 = 0

The functions g and h are such that

h(x) = x² + 4 x 0

What value of x must be excluded from any domain of g?

Solve gh(x) = 1

Curve C has equation y = f (x) The graph of curve C has one maximum point. The coordinates of this maximum point are (3, 5)

Write down the coordinates of the maximum point on the curve with equation

y = 2f (x)

y = f (x) – 7

y = f (–x)

Curve L has equation y = x² + 7x + 20

Find an equation for S Give your answer in the form y = ax² + bx + c

Given that

2ⁿ = 2^x2 16^x × 8

and

x 0

find an expression for x in terms of n State any restrictions on n

Factorise fully 18c³d² – 21c²

Factorise y² – 3y – 18

Hence, solve y² – 3y – 18 = 0

Factorise fully 18c³d² – 21c²

Factorise y² – 3y – 18

Hence, solve y² – 3y – 18 = 0

22Solve the inequality 6x² + 37x 35 Show clear algebraic working.

AEC and BED are chords of a circle.

AE = (x + 5) cm BE = xcm CE = (5x – 12) cm DE = (x + 12) cm

Angle DAE = 48°

Work out the size of angle ADE Give your answer correct to one decimal place.

Expand 3c³(c + 4)

Factorise x² + 8x – 9

Hence, solve x² + 8x –9 = 0

OAB is a sector S of a circle with centre O and radius (r + 7) metres. Angle AOB = 45°

A circle C has radius (r − 2) metres.

The area of sector S is twice the area of circle C

Find the value of r Show your working clearly.

23A polygon has n sides, where n 5

When arranged in order of size, starting with the largest number, the sizes of the interior angles of the polygon, in degrees, are the terms of an arithmetic sequence.

Here are the first five terms of this sequence.

177 175 173 171 169

Find the value of n Show clear algebraic working.

Elliot has x counters.

Each counter has one red face and one green face.

Elliot spreads all the counters out on a table and sees that the number of counters showing a red face is 5

Elliot then picks at random one of the counters and turns the counter over. He then picks at random a second counter and turns the counter over. The probability that there are still 5 counters showing a red face is

Work out the value of x Show clear algebraic working.

19f(x) = x² – 4

g(x) = 2x + 1

Solve fg(x) 0 Show clear algebraic working.

Solve the inequality 5x – 7 ≤ 2

Factorise y² – 2y – 35

Hence, solve y² – 2y – 35 = 0

Solve the inequality 5x – 7 ≤ 2

Factorise y² – 2y – 35

Hence, solve y² – 2y – 35 = 0

Here is a rectangle.

(2 x + 3) cm
Diagram NOT

Given that the area of the rectangle is less than 75 cm²

find the range of possible values of x

Simplify (2x³y⁵)⁴

Factorise x² + 5x – 36

Hence, solve x² + 5x – 36 = 0

Simplify (2x³y⁵)⁴

Factorise x² + 5x – 36

Hence, solve x² + 5x – 36 = 0

Solve x² – 21x + 20 = 0 Show your working clearly.

Solve the equation

Show clear algebraic working.

A bag contains n beads. 6 of the beads are red and the rest are blue.

Ravi is going to take at random 2 beads from the bag. The probability that the 2 beads will be of the same colour is

Using algebra, and showing each stage of your working, calculate the value of n.

Solve 5(4 – x) = 7 – 3 x Show clear algebraic working.

Factorise fully 16m³g³ + 24m²g⁵

Factorise y² – 2y – 48

Hence, solve y² – 2y – 48 = 0

Solve 5(4 – x) = 7 – 3x Show clear algebraic working.

Factorise fully 16m³g³ + 24m²g⁵

Factorise y² – 2y – 48

Hence, solve y² – 2y – 48 = 0

find the values of n for which M = 2

In a bag, there are only

3 blue beads 4 white beads and x orange beads.

Jean is going to take at random two beads from the bag. The probability that Jean will take two beads of the same colour is Find the total number of beads in the bag. Show clear algebraic working.

The diagram shows a rectangle.

The area of the rectangle is Acm²

Given that A 3x + 27 find the range of possible values for x.

A bag contains X counters.

There are only red counters and blue counters in the bag.

There are 4 more blue counters than red counters in the bag.

Finty takes at random 2 counters from the bag. The probability that Finty takes 2 blue counters from the bag is Work out the value of X. Show clear algebraic working.

Solve x²− 5x− 36 = 0 Show clear algebraic working.

the coordinates of any points of intersection of C with the coordinate axes,

the coordinates of the turning point.

A particle moves along a straight line. The fixed point O lies on this line. The displacement of the particle from O at time t seconds, t 0, is s metres where

At time T seconds the velocity of P is V m/s where V −5 Find an expression for T in terms of V.

Solve the inequality 2x + 7 4

Solve x² – 3x – 40 = 0 Show clear algebraic working.

Show clear algebraic working.

Solve the inequality 5y² – 17y 40

Boris has a bag that only contains red sweets and green sweets.

Boris takes at random 2 sweets from the bag. The probability that Boris takes exactly 1 red sweet from the bag is Originally there were 3 red sweets in the bag.

Work out how many green sweets there were originally in the bag. Show your working clearly.

A box contains marbles.

4 of the marbles are red. The rest of the marbles are yellow.

Antonia takes at random a marble from the box and does not replace it. Sergio then takes at random a marble from the box.

The probability that Antonia and Sergio both take a yellow marble is 0.7

Work out how many marbles were originally in the box. Show your working clearly.

Simplify y⁵ × y⁹

Simplify (2m³)⁴

Solve 5(x + 3) = 3x Show clear algebraic working.

Factorise x² + 2x

Hence, solve x² + 2x

Simplify y⁵y⁹

Simplify (2m³)⁴

Solve 5(x + 3) = 3 x – 4 Show clear algebraic working.

Factorise x² + 2x – 24

Hence, solve x² + 2x – 24 = 0

Expand and simplify (2 x – 1)(x + 3)(x – 5)

Solve 3x² + 6x – 5 = 0 Show your working clearly. Give your solutions correct to 3 significant figures.

The points B, C, Y and X lie on a circle.

AXY and ABC are straight lines.

AX = 12 cm XY = x cm AB = x cm BC = 4 cm

Show that x²x

Find the length of AC. Show your working clearly. Give your answer correct to 3 significant figures.

Solve the inequality 4x² – 5x – 6 0

Express 7 – 4x – x² 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

The curve C has equation y = 3 – 5(x + 1)² The point A is the maximum point on C.

Write down the coordinates of A.