Quadratics: Discriminant

$D = b^{2} - 4 a c$

$D > 0$ , two distinct real roots

$D = 0$ , one real root/two coincident roots/repeated root

$D < 0$ , no real root (imaginary roots)

Example

Given quadratic equation $2 x^{2} - 5 x - 3 = 0$

A. Determine the discriminant

B. State the nature of its roots

A. Discriminant

$\begin{aligned} D & = b^{2} - 4 a c \\ D & = (- 5)^{2} - 4 (2) (- 3) \\ D & = 49 \end{aligned}$

B. Nature of roots

Since $D > 0$ , the equation has two distinct real roots.

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$a^{2} + b x + c > 0$

Condition:

$a > 0$ and $D < 0$

Since the curve is concave up, then $a > 0$

Since the curve does not cut x axis, then $D < 0$

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$a^{2} + b x + c < 0$

Condition:

$a < 0$ and $D < 0$

Since the curve is concave down, then $a < 0$

Since the curve does not cut x axis, then $D < 0$

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