TMUA Quadratics and the Discriminant

The discriminant

The discriminant is $b^2-4ac$. Positive means two distinct real roots, zero means one repeated root, and negative means no real roots.

Completing the square

Write $ax^2+bx+c$ as $a(x+p)^2+q$. The vertex is $(-p,q)$, and $q$ is the minimum value when $a>0$ (the maximum when $a<0$).

Sum and product of roots

If $x^2+bx+c=0$ has roots $\alpha$ and $\beta$, then $\alpha+\beta=-b$ and $\alpha\beta=c$. This lets you evaluate symmetric expressions without solving.

Quick checks

The parabola $y=ax^2+bx+c$ opens upwards if $a>0$ and downwards if $a<0$. Its axis of symmetry is $x=-\dfrac b{2a}$, and the two roots are symmetric about it.

If the coefficients are whole numbers and the discriminant is a perfect square, the roots are rational and the quadratic factorises.

Worked examples

Common mistakes

• Treating a zero discriminant as 'no roots': it gives one repeated real root.
• Dropping the factor $a$ when completing the square, so the constant is wrong.
• Using $b^2-4ac$ before the equation is rearranged to equal zero.
• Giving the $x$-value of the vertex when the question asks for the minimum value.

Test yourself on this topic with our free practice questions.