# 10.6: The Discriminant

**At Grade**Created by: CK-12

## Learning Objectives

- Find the discriminant of a quadratic equation.
- Interpret the discriminant of a quadratic equation.
- Solve real-world problems using quadratic functions and interpreting the discriminant.

## Introduction

The quadratic equation is \begin{align*}ax^2 + bx + c = 0\end{align*}.

It can be solved using the quadratic formula \begin{align*} x = \frac{-b \pm \sqrt{b^2 - 4ac}} {2a}\end{align*}.

The expression inside the square root is called the **discriminant**, \begin{align*}D = b^2 - 4ac\end{align*}. The discriminant can be used to analyze the types of solutions of quadratic equations without actually solving the equation. Here are some guidelines.

- If \begin{align*}b^2 - 4ac > 0\end{align*}, we obtain two separate real solutions.
- If \begin{align*}b^2 - 4ac < 0\end{align*}, we obtain non-real solutions.
- If \begin{align*}b^2 - 4ac = 0\end{align*}, we obtain one real solution, a
**double root.**

## Find the Discriminant of a Quadratic Equation

To find the discriminant of a quadratic equation, we calculate \begin{align*}D = b^2 - 4ac\end{align*}.

**Example 1**

*Find the discriminant of each quadratic equation. Then tell how many solutions there will be to the quadratic equation without solving.*

a) \begin{align*}x^2 - 5x + 3 = 0\end{align*}

b) \begin{align*}4x^2 - 4x + 1 = 0\end{align*}

c) \begin{align*}-2x^2 + x = 4\end{align*}

**Solution:**

a) Substitute \begin{align*}a = 1, b = -5\end{align*} and \begin{align*}c = 3\end{align*} into the discriminant formula \begin{align*}D = (-5)^2 - 4(1)(3) = 13\end{align*}.

There are two real solutions because \begin{align*}D > 0\end{align*}.

b) Substitute \begin{align*}a = 4, b = -4\end{align*} and \begin{align*}c = 1\end{align*} into the discriminant formula \begin{align*}D = (-4)^2 - 4(4)(1) = 0\end{align*}.

There is one real solution because \begin{align*}D = 0\end{align*}.

c) Rewrite the equation in standard form \begin{align*}-2x^2 + x - 4 = 0\end{align*}.

Substitute \begin{align*}a = -2, b = 1\end{align*} and \begin{align*}c = -4\end{align*} into the discriminant formula: \begin{align*}D= (1)^2 - 4 (-2) (-4) = -31\end{align*}.

There are no real solutions because \begin{align*}D < 0\end{align*}.

## Interpret the Discriminant of a Quadratic Equation

The sign of the discriminant tells us the nature of the solutions (or roots) of a quadratic equation. We can obtain two distinct real solutions if \begin{align*}D> 0\end{align*}, no real solutions if \begin{align*}D < 0\end{align*} or one solution (called a “double root”) if \begin{align*}D = 0\end{align*}. Recall that the number of solutions of a quadratic equation tell us how many times a parabola crosses the \begin{align*}x-\end{align*}axis.

**Example 2**

*Determine the nature of solutions of each quadratic equation.*

a) \begin{align*}4x^2 - 1 = 0\end{align*}

b) \begin{align*}10x^2 -3x = -4\end{align*}

c) \begin{align*}x^2 - 10x + 25 = 0\end{align*}

**Solution**

Use the value of the discriminant to determine the nature of the solutions to the quadratic equation.

a) Substitute \begin{align*}a = 4, b = 0\end{align*} and \begin{align*}c = -1\end{align*} into the discriminant formula \begin{align*}D = (0)^2 -4(4) (-1) = 16\end{align*}.

The discriminant is positive, so the equation has two distinct real solutions.

The solutions to the equation are: \begin{align*} \frac{0 \pm \sqrt{16}} {8} = \pm \frac{4} {8} = \pm \frac{1} {2}\end{align*}.

b) Rewrite the equation in standard form \begin{align*}10x^2 - 3x + 4 = 0\end{align*}.

Substitute \begin{align*}a = 10, b = -3\end{align*} and \begin{align*}c = 4\end{align*} into the discriminant formula \begin{align*}D = (-3)^2 - 4(10) (4) = -151\end{align*}.

The discriminant is negative, so the equation has two non-real solutions.

c) Substitute \begin{align*}a = 1, b = -10\end{align*} and \begin{align*}c = 25\end{align*} into the discriminant formula \begin{align*}D = (-10)^2 - 4(1) (25) = 0\end{align*}.

The discriminant is \begin{align*}0\end{align*}, so the equation has a double root.

The solution to the equation is \begin{align*} \frac{10 \pm \sqrt{0}} {2} = \frac{10} {2} = 5\end{align*}.

If the discriminant is a perfect square, then the solutions to the equation are rational numbers.

**Example 3**

*Determine the nature of the solutions to each quadratic equation.*

a) \begin{align*}2x^2 + x - 3 = 0\end{align*}

b) \begin{align*}-x^2 - 5x + 14 = 0\end{align*}

**Solution**

Use the discriminant to determine the nature of the solutions.

a) Substitute \begin{align*}a = 2, b = 1\end{align*} and \begin{align*}c = -3\end{align*} into the discriminant formula \begin{align*}D = (1)^2 - 4(2) (-3) = 25\end{align*}.

The discriminant is a positive perfect square so the solutions are two real rational numbers.

The solutions to the equation are \begin{align*} \frac{-1 \pm \sqrt{25}} {4} = \frac{-1 \pm 5} {4}\end{align*} so, \begin{align*}x = 1\end{align*} and \begin{align*} x = -\frac{3} {2}\end{align*}.

b) Substitute \begin{align*}a = -1, b = -5\end{align*} and \begin{align*}c = 14\end{align*} into the discriminant formula: \begin{align*}D = (-5)^2 - 4 (-1)(14) = 81\end{align*}.

The discriminant is a positive perfect square so the solutions are two real rational numbers.

The solutions to the equation are \begin{align*} \frac{5 \pm \sqrt{81}} {-2} = \frac{5 \pm 9} {-2}\end{align*} so, \begin{align*}x = -7\end{align*} and \begin{align*}x = 2\end{align*}.

If the discriminant is not a perfect square, then the solutions to the equation are irrational numbers.

**Example 4**

*Determine the nature of the solutions to each quadratic equation.*

a) \begin{align*}-3x^2 + 4x + 1 = 0\end{align*}

b) \begin{align*}5x^2 - x - 1 = 0\end{align*}

**Solution**

Use the discriminant to determine the nature of the solutions.

a) Substitute \begin{align*}a = -3, b = 2\end{align*} and \begin{align*}c = 1\end{align*} into the discriminant formula \begin{align*}D = (4)^2 - 4(-3)(1) = 28\end{align*}.

The discriminant is a positive perfect square, so the solutions are two real irrational numbers.

The solutions to the equation are \begin{align*} \frac{-2 \pm \sqrt{28}} {-6}\end{align*} so, \begin{align*}x \approx - 0.55\end{align*} and \begin{align*}x \approx 1.22\end{align*}.

b) Substitute \begin{align*}a = 5, b = -1\end{align*} and \begin{align*}c = -1\end{align*} into the discriminant formula \begin{align*}D = (-1)^2 - 4(5) (-1) = 21\end{align*}.

The discriminant is a positive perfect square so the solutions are two real irrational numbers.

The solutions to the equation are \begin{align*} \frac{1 \pm \sqrt{20}} {10}\end{align*} so, \begin{align*}x \approx 0.56\end{align*} and \begin{align*}x \approx -0.36\end{align*}.

## Solve Real-World Problems Using Quadratic Functions and Interpreting the Discriminant

You saw that calculating the discriminant shows what types of solutions a quadratic equation possesses. Knowing the types of solutions is very useful in applied problems. Consider the following situation.

**Example 5**

*Marcus kicks a football in order to score a field goal. The height of the ball is given by the equation* \begin{align*} y = - \frac{32} {6400} x^2 + x\end{align*} *where \begin{align*}y\end{align*} is the height and \begin{align*}x\end{align*} is the horizontal distance the ball travels. We want to know if he kicked the ball hard enough to go over the goal post which is 10 feet high.*

**Solution**

**Define**

Let \begin{align*}y \ =\end{align*} height of the ball in feet

\begin{align*}x \ =\end{align*} distance from the ball to the goalpost.

**Translate** We want to know if it is possible for the height of the ball to equal 10 feet at some real distance from the goalpost.

\begin{align*} 10 = - \frac{32} {6400} x^2 + x\end{align*}

**Solve**

\begin{align*}\text{Write the equation in standard form}. &&& -\frac{32} {6400} x^2 + x - 10 = 0\\ \text{Simplify}. &&& -0.005x^2 + x - 10 = 0\\ \text{Find the discriminant}. &&& D = (1)^2 - 4 (-0.005) (-10)= 0.8\end{align*}

Since the discriminant is positive, we know that it is possible for the ball to go over the goal post, if Marcus kicks it from an acceptable distance \begin{align*}x\end{align*} from the goal post. From what distance can he score a field goal? See the next example.

**Example 6 (continuation)**

*What is the farthest distance that he can kick the ball from and still make it over the goal post?*

**Solution**

We need to solve for the value of \begin{align*}x\end{align*} by using the quadratic formula.

\begin{align*} x = \frac{-1 \pm \sqrt{0.8}} {-0.01} \approx 10.6\ \text{or}\ 189.4\end{align*}

This means that Marcus has to be closer that 189.4 feet or further than 10.6 feet to make the goal. (Why are there two solutions to this equation? Think about the path of a ball after it is kicked).

**Example 7**

*Emma and Bradon own a factory that produces bike helmets. Their accountant says that their profit per year is given by the function*

\begin{align*}P = 0.003x^2 + 12x + 27760\end{align*}

*In this equation \begin{align*}x\end{align*} is the number of helmets produced. Their goal is to make a profit of $40,000 this year. Is this possible?*

**Solution**

We want to know if it is possible for the profit to equal $40,000.

\begin{align*}40000 = -0.003x^2 + 12x + 27760\end{align*}

Solve

\begin{align*}& \text{Write the equation in standard form} & & -0.003x^2 + 12x - 12240 = 0\\ & \text{Find the discriminant}. & & D = (12)^2 - 4(-0.003) (-12240) = -2.88\end{align*}

Since the discriminant is negative, we know that there are no real solutions to this equation. Thus, it is not possible for Emma and Bradon to make a profit of $40,000 this year no matter how many helmets they make.

## Review Questions

Find the discriminant of each quadratic equation.

- \begin{align*}2x^2 - 4x + 5 = 0\end{align*}
- \begin{align*}x^2 - 5x = 8\end{align*}
- \begin{align*}4x^2 - 12x + 9 = 0\end{align*}
- \begin{align*}x^2 + 3x + 2 = 0\end{align*}
- \begin{align*}x^2 - 16x = 32\end{align*}
- \begin{align*}-5x^2 + 5x - 6 = 0\end{align*}

Determine the nature of the solutions of each quadratic equation.

- \begin{align*}-x^2 + 3x - 6 = 0\end{align*}
- \begin{align*}5x^2 = 6x\end{align*}
- \begin{align*}41x^2 - 31x - 52 = 0\end{align*}
- \begin{align*}x^2 - 8x + 16 = 0\end{align*}
- \begin{align*}-x^2 + 3x - 10 = 0\end{align*}
- \begin{align*}x^2 - 64 = 0\end{align*}

Without solving the equation, determine whether the solutions will be rational or irrational.

- \begin{align*}x^2 = - 4x + 20\end{align*}
- \begin{align*}x^2 + 2x - 3 = 0\end{align*}
- \begin{align*}3x^2 - 11x = 10\end{align*}
- \begin{align*} \frac{1} {2} x^2 + 2x + \frac{2}{3} = 0\end{align*}
- \begin{align*}x^2 - 10x + 25 = 0\end{align*}
- \begin{align*}x^2 = 5x\end{align*}
- Marty is outside his apartment building. He needs to give Yolanda her cell phone but he does not have time to run upstairs to the third floor to give it to her. He throws it straight up with a vertical velocity of 55 feet/second. Will the phone reach her if she is 36 feet up? (Hint: The equation for the height is given by \begin{align*}y = -32t^2 + 55t + 4\end{align*}.)
- Bryson owns a business that manufactures and sells tires. The revenue from selling the tires in the month of July is given by the function \begin{align*}R = x(200 - 0.4x)\end{align*} where \begin{align*}x\end{align*} is the number of tires sold. Can Bryson’s business generate revenue of $20,000 in the month of July?

## Review Answers

- \begin{align*}D = -24\end{align*}
- \begin{align*}D = 57\end{align*}
- \begin{align*}D = 0\end{align*}
- \begin{align*}D = 1\end{align*}
- \begin{align*}D = 384\end{align*}
- \begin{align*}D = -95\end{align*}
- \begin{align*}D = -15\end{align*} no real solutions
- \begin{align*}D = 36\end{align*} two real solutions
- \begin{align*}D = 9489\end{align*} two real solutions
- \begin{align*}D = 0\end{align*} one real solutions
- \begin{align*}D = -31\end{align*} no real solutions
- \begin{align*}D = 256\end{align*} two real solutions
- \begin{align*}D = 96\end{align*} two real irrational solutions
- \begin{align*}D = 16\end{align*} two real rational solutions
- \begin{align*}D = 241\end{align*} two real irrational solutions
- \begin{align*}D = \frac{8}{3}\end{align*} two real irrational solutions
- \begin{align*}D = 0\end{align*} one real rational solution
- \begin{align*}D = 25\end{align*} two real rational solutions
- no
- yes