GCSE Maths Quadratic equations & graphs questions by topic

Solve by factorising.

1. (a)Solve x² + 7x + 12 = 0.[2]
2. (b)Solve x² − 4x − 21 = 0.[2]

Reading a parabola.

1. (a)y = (x − 1)(x + 4). Write down the roots of the equation y = 0.[2]
2. (b)Write down the coordinates of the y-intercept.[1]
3. (c)State whether the parabola opens upwards or downwards, and why.[1]

Rearrange then solve.

1. (a)Solve x² = 8x + 20.[3]
2. (b)Check the positive root by substitution.[1]

Difference of two squares.

1. (a)Solve x² − 25 = 0.[2]
2. (b)Solve (2x − 3)(2x + 3) = 0 and state the sum of your two roots.[2]

A rectangular garden has an unknown width.

1. (a)The garden is x metres wide and (x + 3) metres long, with area 40 m². Show that x² + 3x = 40.[1]
2. (b)Solve the equation in part (a) by factorising.[3]
3. (c)Write down the width of the garden.[1]

A model rocket is launched.

1. (a)Its height in metres is h = 24t − 6t² after t seconds. Factorise 24t − 6t² completely.[1]
2. (b)Work out the time at which the rocket lands.[2]
3. (c)The rocket is highest halfway through the flight. Work out its greatest height.[2]

The quadratic formula. (Higher)

1. (a)Write down the quadratic formula.[1]
2. (b)Use it to solve 3x² + 2x − 6 = 0, giving roots to 2 d.p.[4]

Exact roots. (Higher)

1. (a)Solve x² − 6x + 2 = 0, giving your answers in the form p ± √q.[3]
2. (b)Write down the sum of the roots.[1]

Completing the square. (Higher)

1. (a)Write x² + 8x + 3 in the form (x + p)² + q.[2]
2. (b)Hence solve x² + 8x + 3 = 0, giving roots in surd form.[2]
3. (c)State the minimum point of y = x² + 8x + 3.[1]

Graphs and roots.

1. (a)The graph of y = x² − 5x + 4 crosses the x-axis at P and Q. Find their coordinates.[3]
2. (b)Write down the equation of the axis of symmetry.[1]

The quadratic formula is applied step by step. (Higher)

1. (a)For 2x² − 7x + 3 = 0, write down the values of a, b and c.[1]
2. (b)Work out b² − 4ac.[1]
3. (c)Hence solve the equation.[3]

Turning points from the completed square. (Higher)

1. (a)Write 2x² − 12x + 5 in the form a(x − p)² + q. (Factorise the 2 out first.)[3]
2. (b)State the coordinates of the turning point of y = 2x² − 12x + 5.[2]
3. (c)Write down the minimum value of y.[1]

Formula with rearrangement. (Higher)

1. (a)Show that 2x² = 4x + 5 can be written as 2x² − 4x − 5 = 0.[1]
2. (b)Solve 2x² − 4x − 5 = 0 using the formula, giving roots to 2 d.p.[4]
3. (c)Explain why there are two real roots.[1]

A discriminant investigation. (Higher)

1. (a)Find the discriminant of x² − 6x + k and write it in terms of k.[2]
2. (b)For which values of k does the equation have two distinct real roots?[2]
3. (c)What happens at k = 9?[2]

From roots to an equation. (Higher)

1. (a)A parabola has roots 4 and −1 and a y-intercept of −8. Find its equation in the form y = ax² + bx + c.[5]
2. (b)Find the equation of its axis of symmetry.[1]
3. (c)Does the parabola open upwards or downwards?[1]

A parabola is rewritten and used. (Higher)

1. (a)Express y = x² − 10x + 18 in completed-square form.[3]
2. (b)Hence state the pair (x, y) at which the curve bottoms out.[1]
3. (c)Hence solve x² − 10x + 18 = 0, giving exact answers.[2]

A quadratic with an unknown coefficient. (Higher)

1. (a)The equation kx² + 6x + 2 = 0 has two equal roots. Write down the condition this places on the discriminant.[1]
2. (b)Find k.[2]
3. (c)For your value of k, solve the equation.[3]

Rewriting 2x² − 12x + 5. (Higher)

1. (a)Write 2x² − 12x + 5 in the form a(x + p)² + q.[3]
2. (b)Write down the minimum value of 2x² − 12x + 5.[1]
3. (c)Hence solve 2x² − 12x + 5 = 0, giving your answers to 2 decimal places.[3]