Theory of Intelligence
A-LEVEL•Mathematics•Quadratics•Question 1

Solution

🟢 Foundation · [3 marks] Express $2x^2 - 12x + 23$ in the form $a(x+b)^2 + c$, and state the coordinates of the vertex of $y = 2x^2 - 12x + 23$.

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Solution

2x2−12x+23=2[x2−6x]+23=2[(x−3)2−9]+23=2(x−3)2−18+23=2(x−3)2+5\begin{aligned} 2x^2 - 12x + 23 &= 2\left[x^2 - 6x\right] + 23 \\ &= 2\left[(x-3)^2 - 9\right] + 23 \\ &= 2(x-3)^2 - 18 + 23 \\ &= 2(x-3)^2 + 5 \end{aligned}

So a=2a = 2, b=−3b = -3, c=5c = 5, and the vertex is (3, 5)(3,\ 5), which is a minimum point.

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