Mathematics (AS)·Equations, expressions, identities and inequalities · NSSCAS 1.1.2
The discriminant & the nature of roots
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Hiding inside the quadratic formula is a tiny expression with a big job: the discriminant, b² − 4ac. In this lesson you will learn to calculate it for any quadratic, and to read from its sign whether the equation has two distinct real roots, one repeated root, or no real roots at all — before you solve anything! You will also link each case to a picture of the parabola, and master the classic exam question: finding an unknown constant k from the nature of the roots.
What you'll learn in this lesson
By the end you should be able to (NSSCAS Mathematics (AS) 1.1.2):
- Find the discriminant b² − 4ac of a quadratic polynomial ax² + bx + c, reading off a, b and c with their correct signs
- Interpret the nature and number of real roots of ax² + bx + c = 0: two distinct real roots when b² − 4ac > 0, one repeated real root when b² − 4ac = 0, no real roots when b² − 4ac < 0, and link each case to how the parabola meets the x-axis
- Use the discriminant to find unknown constants, for example find k so that kx² + 4x + 1 = 0 has equal roots, or find the range of values of k for which an equation has no real roots
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Miss Eve and Mike talk through the whole topic — with the figure and working drawn live.