20 (a) Write (2x − 5)(x + 4) in the form 2(x + a)² − b - OCR - GCSE Maths - Question 20 - 2023 - Paper 6

Next, we factor out the coefficient of $x^2$, which is 2: $2(x^2 + \frac{3}{2}x - 10)$

To write the expression in the desired form, complete the square for the quadratic expression within the parentheses:

The term to complete the square is igg(\frac{b}{2}\bigg)^2 = \bigg(\frac{\frac{3}{2}}{2}\bigg)^2 = \bigg(\frac{3}{4}\bigg)^2 = \frac{9}{16}.

Thus: $(x^2 + \frac{3}{2}x) = (x + \frac{3}{4})^2 - \frac{9}{16}$

Substituting this back in gives us: $2\bigg((x + \frac{3}{4})^2 - \frac{9}{16} - 10\bigg)$

Now simplifying further:

$2\bigg((x + \frac{3}{4})^2 - \frac{169}{16}\bigg)$

Hence, we rewrite this as: $2(x + \frac{3}{4})^2 - \frac{338}{16}$

We can express this in the form $2(x + a)^2 - b$, where:

• $a = \frac{3}{4}$
• $b = \frac{338}{16}$