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 2 from the quadratic part:

2x^2 + 3x - 20 & = 2(x^2 + \frac{3}{2}x) - 20.\end{aligned}$$

Now, we will complete the square for the expression inside the parentheses:

1. Take half of the coefficient of x, which is $\frac{3}{4}$, and square it to get $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$.
2. Add and subtract this square inside the parentheses:

$\begin{aligned}2\left(x^2 + \frac{3}{2}x + \frac{9}{16} - \frac{9}{16}\right) - 20&= 2\left((x + \frac{3}{4})^2 - \frac{9}{16}\right) - 20.\end{aligned}$

Distribute the 2 and simplify:

& = 2(x + \frac{3}{4})^2 - \frac{169}{8}. \end{aligned}$$ Thus, we can express it in the required form $2(x + a)^2 - b$ where $a = \frac{3}{4}$ and $b = \frac{169}{8}$.