A household is trying to bring its finances back under control after an expensive winter, and its monthly budget shortfall, in pounds, satisfies \(x^2 - 6x ...
Question 1 Report
A household is trying to bring its finances back under control after an expensive winter, and its monthly budget shortfall, in pounds, satisfies \(x^2 - 6x - 2 = 0\), where \(x\) is the number of streaming subscriptions the family cancels.
Solve \(x^2 - 6x - 2 = 0\) by completing the square, giving your answers in the form \(a \pm \sqrt{b}\). (4)
Answer Details
Completing the square rewrites a quadratic \( x^2+bx+c \) as \( (x+\frac{b}{2})^2 - (\frac{b}{2})^2 + c \), which turns the equation into one that can be solved by taking a square root directly, giving exact surd answers.
For \( x^2-6x-2=0 \), halving the coefficient of \(x\) gives \(-3\), so \( x^2-6x \) can be written as \( (x-3)^2-9 \) (since \( (x-3)^2 = x^2-6x+9 \), the \(-9\) corrects for the extra \(+9\) introduced). Substituting this back gives \( (x-3)^2-9-2=0 \) [1 mark], which simplifies to \( (x-3)^2 = 11 \) [1 mark].
Taking the square root of both sides, and keeping both the positive and negative root since squaring loses the sign, gives \( x-3 = \pm\sqrt{11} \) [1 mark]. Adding 3 to both sides gives \( x = 3 \pm \sqrt{11} \) [1 mark].
Leaving the answer as \( 3 \pm \sqrt{11} \), rather than converting to a decimal, gives the exact solutions requested; \( \sqrt{11} \) is irrational, so no exact decimal exists, and rounding it early would lose the precision the question asks for.
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