Solve the equation, giving your answers correct to 2 decimal places. \(\frac{5}{x} + \frac{3}{x+2} = 2\)

Fractions with \(x\) in the denominator are cleared by multiplying every term by the product of the denominators, here \(x(x+2)\). That turns the equation into a quadratic.

Multiplying \(\dfrac{5}{x} + \dfrac{3}{x+2} = 2\) through by \(x(x+2)\):

\(5(x+2) + 3x = 2x(x+2)\) [M1]

Expand each side: \(5x + 10 + 3x = 2x^2 + 4x\), that is \(8x + 10 = 2x^2 + 4x\). Collect everything on one side so the quadratic equals zero:

\(2x^2 - 4x - 10 = 0\) oe, which divides by 2 to give \(x^2 - 2x - 5 = 0\) [M1]

This does not factorise, so use the quadratic formula with \(a = 1\), \(b = -2\), \(c = -5\):

\(x = \dfrac{2 \pm \sqrt{(-2)^2 - 4(1)(-5)}}{2} = \dfrac{2 \pm \sqrt{24}}{2}\)

\(\sqrt{24} = 4.8989\ldots\), so \(x = \dfrac{2 + 4.8989}{2} = 3.4494\ldots\) or \(x = \dfrac{2 - 4.8989}{2} = -1.4494\ldots\)

\(x = 3.45\) and \(x = -1.45\) (2 d.p.) [A2]

[A1 is available for one correct value, or for both values seen to greater accuracy.]

Both roots are valid: neither makes a denominator zero, since \(x \ne 0\) and \(x \ne -2\). Two warnings. First, the instruction "correct to 2 decimal places" signals that the quadratic will not factorise, so reach for the formula rather than hunting for factors. Second, keep the unrounded values in the calculator until the final step; rounding \(\sqrt{24}\) to 4.9 early shifts the second decimal place.