GB-19-02-001 · Non-monic quadratic equations

Solve $8x^{2} - 108x + 52=0$ .

Solving Equations: Quadratics by Factorising (a > 1) Booklet p. 6 & p. 12

Method

1) Rearrange = 0

2) Factorise using a × c and 2×2 grid

3) Put each bracket = 0 and solve

Worked Example (GB-19-02-001)

① Booklet p. 4 — Take out the numerical HCF

\mathbf{4}

first:

\quad 8x^2 - 108x + 52 = 4(2x^2 - 27x + 13)

2) Factorise using 2×2 Grid & a×c T-table (Booklet p. 5–6):

×	$2x$	$-1$
$x$	$2x^2$	$-x$
$-13$	$-26x$	$+13$

a \times c = 2 \times (13) = 26

$-1$	$-26$
$-2$	$-13$

c is positive so both signs are the same (− and −)...

-1 - 26 = -27

✔

↳ Need to take the HCF out as factor: here

\mathbf{x}

is the HCF of

2x^2

and

-x

\text{Brackets: } \mathbf{(x - 13)(2x - 1)}

↳ The HCF

4

can never equal

0

, so only the brackets give solutions.

3) Put each bracket = 0 and solve (Booklet p. 12):

x - 13 = 0

+13

+13

\mathbf{x = 13}

2x - 1 = 0

+1

+1

2x = 1

\div 2

\div 2

\mathbf{x = \frac{1}{2}}

x=\frac{1}{2}

x=13