Simplify $\frac{5a^3}{6b} + \frac{2ab}{4a^2b}$. Solve $\frac{x}{8} = \frac{50}{x}$. Simplify $\frac{4v^2 - 1}{2pv + p - 10v - 5}$. Solve $2x^2 = 3(3x - 1).$ Give your solutions correct to two decimal places. Answer: $x =$ ... or ...

1. Simplify the individual fraction and add using a common denominator

$\frac{5a^3}{6b}+\frac{2ab}{4a^2b}$

$=\frac{5a^3}{6b}+\frac{1}{2a}$

Common denominator: $6ab$.

$\frac{5a^3}{6b}=\frac{5a^4}{6ab}, \qquad \frac{1}{2a}=\frac{3b}{6ab}$

$\frac{5a^4}{6ab}+\frac{3b}{6ab}=\frac{5a^4+3b}{6ab}$

1. Clear the fractions to form a quadratic equation

$\frac{x}{8}=\frac{50}{x}$
Since $x\ne 0$, multiply both sides by $8x$:
$x^2=400$

2. Solve the quadratic equation

$x^2=400$
$x=\pm\sqrt{400}$
$x=20 \text{ or } x=-20$

1. Factorise the numerator using difference of squares

$4v^2-1=(2v)^2-1^2=(2v+1)(2v-1)$

2. Factorise the denominator by grouping

3. Cancel the common factor

$\frac{4v^2-1}{2pv+p-10v-5}=\frac{(2v+1)(2v-1)}{(2v+1)(p-5)}=\frac{2v-1}{p-5}$

1. Expand the brackets and rearrange into standard quadratic form

$2x^2=3(3x-1)$
$2x^2=9x-3$
$2x^2-9x+3=0$

2. Apply the quadratic formula

For $2x^2-9x+3=0$, $a=2$, $b=-9$, $c=3$.

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\frac{-(-9)\pm\sqrt{(-9)^2-4(2)(3)}}{2(2)}$
$x=\frac{9\pm\sqrt{57}}{4}$

3. Evaluate both roots and round to two decimal places

$x=\frac{9+\sqrt{57}}{4}=4.137\ldots \approx 4.15$

$x=\frac{9-\sqrt{57}}{4}=0.362\ldots \approx 0.35$