Factorise \( x^2 + 5x + 6 \).
\( (x+2)(x+3) \).
Quadratic Equations
10 exam-style questions on factorising, the quadratic formula and completing the square (GCSE level).
3 Easy 4 Medium 3 Hard
Solve \( x^2 - 9 = 0 \).
\( x^2 = 9 \) ⇒ \( x = \pm 3 \).
Solve \( x^2 - 5x + 6 = 0 \) by factorising.
\( (x-2)(x-3) = 0 \) ⇒ \( x = 2 \) or \( x = 3 \).
Solve \( x^2 + 4x - 12 = 0 \) by factorising.
\( (x+6)(x-2) = 0 \) ⇒ \( x = -6 \) or \( x = 2 \).
Use the quadratic formula to solve \( x^2 + 3x - 5 = 0 \). Leave answers in surd form.
\( x = \dfrac{-3 \pm \sqrt{9+20}}{2} = \dfrac{-3 \pm \sqrt{29}}{2} \).
Complete the square for \( x^2 + 6x + 5 \) and hence find the minimum value.
\( (x+3)^2 - 9 + 5 = (x+3)^2 - 4 \). Minimum \( = -4 \) at \( x = -3 \).
Solve \( 2x^2 - 5x - 3 = 0 \) by factorising.
\( (2x + 1)(x - 3) = 0 \) ⇒ \( x = -\dfrac{1}{2} \) or \( x = 3 \).
The equation \( x^2 + kx + 9 = 0 \) has equal roots. Find \( k \).
Discriminant \( = 0 \): \( k^2 - 36 = 0 \) ⇒ \( k = \pm 6 \).
Solve \( x^2 - 4x - 1 = 0 \) by completing the square. Leave answers in surd form.
\( (x-2)^2 - 4 - 1 = 0 \) ⇒ \( (x-2)^2 = 5 \) ⇒ \( x = 2 \pm \sqrt{5} \).
A rectangle has length \( (x+3) \) cm and width \( (x-1) \) cm. Its area is \( 32 \) cm². Form and solve a quadratic equation to find \( x \).
\( (x+3)(x-1) = 32 \) ⇒ \( x^2 + 2x - 3 = 32 \) ⇒ \( x^2 + 2x - 35 = 0 \) ⇒ \( (x+7)(x-5)=0 \). \( x = 5 \) (since length > 0).