Work out the smallest value of a that is a solution to the equation 4/(2a-1) + (a+7)/(3a−5) = 2. Give your answer to 2 d.p.

$$\frac{4(3a - 5) + (a + 7)(2a - 1)}{(2a - 1)(3a - 5)} = 2$$

Expand the terms in the numerator:

$$4(3a - 5) = 12a - 20$$ $$(a + 7)(2a - 1) = 2a^2 - a + 14a - 7 = 2a^2 + 13a - 7$$

Sum of numerators:

$$(12a - 20) + (2a^2 + 13a - 7) = 2a^2 + 25a - 27$$

Expand the terms in the denominator:

$$(2a - 1)(3a - 5) = 6a^2 - 10a - 3a + 5 = 6a^2 - 13a + 5$$

The equation becomes:

$$\frac{2a^2 + 25a - 27}{6a^2 - 13a + 5} = 2$$

$$2a^2 + 25a - 27 = 2(6a^2 - 13a + 5)$$ $$2a^2 + 25a - 27 = 12a^2 - 26a + 10$$

Subtract $2a^2$, $25a$, and $-27$ from both sides:

$$0 = 12a^2 - 2a^2 - 26a - 25a + 10 + 27$$ $$0 = 10a^2 - 51a + 37$$

Here, $A = 10$, $B = -51$, $C = 37$.

$$a = \frac{-(-51) \pm \sqrt{(-51)^2 - 4(10)(37)}}{2(10)}$$ $$a = \frac{51 \pm \sqrt{2601 - 1480}}{20}$$ $$a = \frac{51 \pm \sqrt{1121}}{20}$$

First, calculate the square root: $\sqrt{1121} \approx 33.481338$

$$a_1 = \frac{51 + 33.481338}{20} = \frac{84.481338}{20} \approx 4.2240669$$ $$a_2 = \frac{51 - 33.481338}{20} = \frac{17.518662}{20} \approx 0.8759331$$

Comparing $a_1 \approx 4.22$ and $a_2 \approx 0.88$, the smallest value is 0.8759331.
Rounding to 2 decimal places, this is 0.88.

$$a = 0.88$$