Using the Quadratic Formula and the Discriminant

The Discriminant

For $a x^{2}+b x+c=0$, where $a, b$, and $c$ are real numbers, the discriminant is the expression under the radical in the quadratic formula: $b^{2}-4 a c$. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.

Example 11

Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation

Use the discriminant to find the nature of the solutions to the following quadratic equations:

(a) $x^{2}+4 x+4=0$

(b) $8 x^{2}+14 x+3=0$

(c) $3 x^{2}-5 x-2=0$

(d) $3 x^{2}-10 x+15=0$

Solution

Calculate the discriminant $b^{2}-4 a c$ for each equation and state the expected type of solutions.

(a)

$x^{2}+4 x+4=0$

$b^{2}-4 a c=(4)^{2}-4(1)(4)=0$. There will be one rational double solution.

(b)

$8 x^{2}+14 x+3=0$

$b^{2}-4 a c=(14)^{2}-4(8)(3)=100$. As $100$ is a perfect square, there will be two rational solutions.

(c)

$3 x^{2}-5 x-2=0$

$b^{2}-4 a c=(-5)^{2}-4(3)(-2)=49$. As $49$ is a perfect square, there will be two rational solutions.

(d)

$3 x^{2}-10 x+15=0$

$b^{2}-4 a c=(-10)^{2}-4(3)(15)=-80$. There will be two complex solutions.