M1: for a correct method to find the critical values
Minimum evidence for quadratic formula is a two-term discriminant, eg \(\dfrac{7 \pm \sqrt{49 + 120}}{4}\) (must have the \(\pm\))
Allow \((x + 1.5)(2x - 10)\) as a correct factorisation, but do not allow \(\left(x + \dfrac{3}{2}\right)(x - 5)\) unless preceeded by division of the quadratic by 2
A1: dep on M1 for correct critical values oe
A1: dep on M1
for correct inequalities (must be separate inequalities)
or \(\left]-\infty, -\dfrac{3}{2}\right[ \cup \left]5, \infty\right[\) or \(\left]-\infty, -\dfrac{3}{2}\right[, \left]5, \infty\right[\)
Acceptable notation: allow a comma, space, “or”, “and” or “\(\cup\)” to link the two regions
Do not allow as a single inequality \(-\dfrac{3}{2} \gt x \gt 5\)
Allow M1A1 for the correct critical values AND evidence of another algebraic method that has led to these: eg \(x(2x + 3) - 5(2x + 3)\) and the correct critical values or \(2x(x - 5) + 3(x - 5)\) and the correct critical values or \((2x + 3)(2x - 10)\) and the correct critical values
M1: for a correct method to solve the quadratic equation \(4x^2 + 4x - 15 = 0\)
Allow \((4x - 6)(x + 2.5)\) or \((4x + 10)(x - 1.5)\) or \((4x - 6)(4x + 10)\) leading to \((x - 1.5)(x + 2.5)\) or \((4x - 6)(4x + 10)\) leading to correct values of \(x\)
Do not allow \((x - 1.5)(x + 2.5)\) without previous working
(If using formula allow some simplification – allow as far as \(\dfrac{-4 \pm \sqrt{16 + 240}}{8}\))
A1: oe dep on M1
A1: oe dep on M1 Allow \(x \gt -2.5\) (and) \(x \lt 1.5\) oe Allow any variable as long as used all the way through
M1: A correct method to solve the quadratic; correct factors or correct substitution into the formula or can be simplified as far as \(\dfrac{-11 \pm \sqrt{121 + 840}}{20}\) or correctly completing the square.
\((10)(x + 2.1)(x - 1)\) is not a correct factorisation – it is working backwards from calculator answers.
A1: dep on M1 for correct critical values
A1: dep on M1
oe eg \(x \gt -2.1\) (and) \(x \lt 1\) (do not penalise ‘or’)
or \(\dfrac{-21}{10} \lt x \lt 1\) etc
including the open interval (–2.1, 1) or ]–2.1, 1[
