Quick questions on Algebra: quadratics, the discriminant, simultaneous equations and inequalities - OCR A-Level Maths A

The discriminant is $\Delta = b^2 - 4ac$. It tells you the nature of the roots without solving:

When one equation is linear and one is quadratic, substitute the linear equation into the quadratic to get a single quadratic in one variable, solve it, then back-substitute.

For a linear inequality, solve as an equation, but reverse the sign when multiplying or dividing by a negative. For a quadratic inequality, find the roots, sketch the parabola, and read off the region. An upward parabola is positive outside the roots and negative between them.

Many equations are quadratics in disguise once you substitute. Equations in $x^4$ and $x^2$, in $\sqrt{x}$, or with a repeated bracket all reduce to a quadratic by a single substitution, which you then solve and reverse.

Completing the square does more than solve a quadratic: $a(x + p)^2 + q$ shows the vertex is at $(-p, q)$, the minimum value (for $a > 0$) is $q$, and the line of symmetry is $x = -p$. This is why completing the square, rather than the formula, is the right tool when a question asks for the minimum value of a quadratic or the range of a quadratic function.

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

Solve $x^2 - 2x - 8 \le 0$. [3 marks]