Quadratics

Discriminant

 

Discriminant

\(D = b^2 - 4ac\)

 

\(D > 0\), 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 \(2x^2 - 5x - 3= 0\)

A. Determine the discriminant

B. State the nature of its roots

 

A. Discriminant

\begin{equation*} \begin{split} D & = b^2 - 4ac \\\\ D & = (-5)^2 - 4(2)(-3)\\\\ D & = 49 \end{split} \end{equation*}

 

B. Nature of roots

Since \(D > 0\), the equation has two distinct real roots.

Definite Form
Positive Definite

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\(a^2 + bx + c > 0\)

 

Condition:

\(a > 0\) and \(D < 0\)

Since the curve is concave up, then \(a > 0\)

Since the curve does not cut x axis, then \(D < 0\)

 

 

Negative Definite

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\(a^2 + bx + c < 0\)

 

Condition:

\(a < 0\) and \(D < 0\)

Since the curve is concave down, then \(a < 0\)

Since the curve does not cut x axis, then \(D < 0\)

 

Exercise

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