The discriminant
Watch this video to learn about the nature of roots using the discriminant.
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)
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- Working with linear equations and inequations
- Working with simultaneous equations
- Changing the subject of a formula
- Determine the equation of a quadratic function from its graph
- Sketching a quadratic function
- Identifying features of a quadratic function
- Solving a quadratic equation
- Solving a quadratic equation using the quadratic formula
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