Quick questions on Completing the square explained: O-Level A-Maths

For $x^2 + bx + c$, halve the coefficient of $x$, square it, add and subtract it:

Factor $a$ out of the $x$ terms first, complete the square inside the bracket, then expand the constant back out:

The completed-square form gives everything needed for a quick sketch: plot the vertex $(h, k)$, draw the axis of symmetry through it, note whether the curve opens up or down from the sign of $a$, and mark the $y$-intercept by setting $x = 0$ in the original. The shape follows without a table of values.

Completing the square also solves a quadratic equation: rearranging $a(x - h)^2 + k = 0$ to $(x - h)^2 = -\dfrac{k}{a}$ and taking the square root gives the roots directly. This is the manipulation that derives the quadratic formula itself.

You subtract $\left(\tfrac{b}{2}\right)^2$ inside; multiplying back out by $a$ changes its contribution to the constant.

Express $x^2 + 8x + 3$ in the form $(x + a)^2 + b$. [2 marks]

State the minimum value of $(x - 5)^2 + 7$ and where it occurs. [1 mark]

Express $2x^2 - 12x + 1$ in completed-square form. [3 marks]