Whether the discriminant is greater than zero, equal to zero or less than zero can be used to determine if a quadratic equation has no real roots, real and equal roots or real and unequal roots.

Part of

If \(kx^{2}+5x-\frac{5}{4}=0\) has equal roots, then \(b^2-4ac=0\).

\(a=k\), \(b=5\) and \(c= - \frac{5}{4}\).

\[b^2-4ac=0\]

\[5^2 -4\times k \times - \frac{5}{4}=0\]

\[25+5k=0\]

Rearrange to make \(k\) the subject.

\[5k=-25\]

\[k=-5\]

The discriminant is \({b^2} - 4ac\), which comes from the quadratic formula and we can use this to find the nature of the roots. Roots can occur in a parabola in 3 different ways as shown in the diagram below:

In the first diagram, we can see that this parabola has two roots. The second diagram has one root and the third diagram has no roots.

The discriminant can be used in the following way:

\({b^2} - 4ac\textless0\) - there are no real roots (diagram 1)

\({b^2} - 4ac = 0\) - the roots are real and equal ie one real root (diagram 2)

\({b^2} - 4ac\textgreater0\) - the roots are real and unequal ie two distinct real roots (diagram 3)