Quadratics — common mistakes

Exam questions frequently ask for the set of values of a constant, say $k$, for which a quadratic equation has 'real roots' or 'two distinct roots'. For 'real roots', use the condition $b^2 - 4ac \ge 0$. For 'two distinct roots', use the strict inequality $b^2 - 4ac > 0$. Be careful with the wording!

When completing the square for $ax^2+bx+c$ where $a \neq 1$, always factor out $a$ from the $x^2$ and $x$ terms first. A common error is to forget to multiply the subtracted term by $a$ when taking it out of the bracket.

They are closely related concepts. 'Roots' or 'solutions' refer to the algebraic values of $x$ that solve the equation $ax^2+bx+c=0$. 'x-intercepts' are the geometric points where the graph of the function $y=ax^2+bx+c$ crosses the x-axis. The x-coordinates of these intercepts are the real roots of the equation.

Always try to factorise first, as it's faster. If you can't spot the factors within about 20-30 seconds, or if the question asks for an answer to a specific number of decimal places or significant figures, it's a strong hint that you should use the quadratic formula.

If you have $ax^2+bx+c$, you must first factor out the coefficient 'a' from the first two terms, giving $a(x^2 + (b/a)x) + c$. Then, you complete the square inside the bracket as usual and multiply out at the end. See the second worked example for a step-by-step guide.

Graphically, 'no real roots' means the parabola does not touch or cross the x-axis at all. If the coefficient 'a' is positive, the entire curve will be above the x-axis. If 'a' is negative, the entire curve will be below the x-axis.