Quick questions on Quadratic formula and completing the square explained: O-Level E-Maths

For any quadratic $ax^2 + bx + c = 0$, the solutions are:

The quantity under the root, $b^2 - 4ac$, is the discriminant. If it is positive there are two real solutions, if it is zero there is one repeated solution, and if it is negative there are no real solutions because you cannot take the square root of a negative number at this level.

Completing the square rewrites $x^2 + bx$ as $\left(x + \dfrac{b}{2}\right)^2 - \left(\dfrac{b}{2}\right)^2$. Once the equation is a perfect square equal to a number, take the square root of both sides, remembering the plus and minus, then solve.

Written as $(x + a)^2 + b$, a quadratic has its minimum value $b$ when the squared term is zero, at $x = -a$. This is how completing the square locates the turning point of a parabola without calculus, useful in the graphs strand.

Use the formula to solve $x^2 + 3x - 2 = 0$, giving answers to 2 decimal places. [3 marks]

Express $x^2 - 4x + 7$ in the form $(x - a)^2 + b$. [2 marks]

State the number of real solutions of $x^2 + 2x + 5 = 0$ and justify. [2 marks]