Example 1

\(a = 4, b = 6, c = -10\)

\(\begin{align}
 b^2 - 4ac &= \left( 6 \right)^2 - 4\left( 4 \right)\left( { -10} \right) \\
  &= 36 + 160 \\
  &= 196 \\ 
\end{align}\)

Because \(b^2 - 4ac > 0\), the quadratic equation \(4x^2 + 6x - 10 = 0\) has two distinct Real roots. Graphing the corresponding quadratic function, as shown in the screen capture, verifies the nature of the roots.

Note: Before you can determine the discriminant, you must change to the standard form \(ax^2 + bx + c = 0\).

\(\begin{array}{l}
 9x^2 - 12x = -4 \\
 9x^2 - 12x + 4 = 0 \\
 \end{array}\)

\(a = 9, b = -12, c = 4\)

\(\begin{align}
 b^2 - 4ac &= \left( { -12} \right)^2 - 4\left( 9 \right)\left( 4 \right) \\
  &= 144 - 144 \\
  &= 0 \\
 \end{align}\)

Because \(b^2 - 4ac = 0\), there are two Real and equal roots. Graphing the corresponding quadratic function, as shown in the screen capture, verifies the nature of the roots.

\(a = 3, b = -4, c = 5\)

\(\begin{align}
 b^2 - 4ac &= \left( { - 4} \right)^2 - 4\left( 3 \right)\left( 5 \right) \\
  &= 16 - 60 \\
  &= -44 \\
 \end{align}\)

Because \(b^2 - 4ac < 0\), there are zero Real roots and two imaginary roots, which are complex numbers. Graphing the corresponding quadratic function, as shown in the screen capture, verifies the nature of the roots.