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Quadratic Equations — Part 2: Nature of roots and word problems

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Lesson

Quadratic Equations, Part 2: Nature of roots and applications

One number, $b^2 - 4ac$ , tells you how many roots to expect.

In this lesson you will

Find the discriminant b² − 4ac and state the nature of the roots
Find unknown coefficients for equal or real roots
Formulate quadratic equations from area, number, speed and other situations
Solve them and reject inadmissible roots
Use the discriminant to decide whether a situation is possible

$D > 0$
$D = 0$
$D < 0$

The discriminant predicts the picture: two crossings, a touch, or none.

Diagram

The discriminant and the nature of roots

Nature of roots from the discriminant.
$D = b^2 - 4ac$	Roots	Graph of $y = ax^2 + bx + c$
$D > 0$	two distinct real roots	crosses the $x$ -axis twice
$D = 0$	two equal real roots, $-\dfrac{b}{2a}$	touches the $x$ -axis once
$D < 0$	no real roots	does not meet the $x$ -axis

In the formula, $\sqrt{b^2 - 4ac}$ decides everything. Call $D = b^2 - 4ac$ the discriminant.

D = b^2 - 4ac