### Solutions Using the Discriminant

In 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.** The discriminant can be used to analyze the types of solutions to a quadratic equation without actually solving the equation. Here’s how:

- If \begin{align*}b^2-4ac>0\end{align*}, the equation has two separate real solutions.
- If \begin{align*}b^2-4ac<0\end{align*}, the equation has only non-real solutions.
- If \begin{align*}b^2-4ac=0\end{align*}, the equation has 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*}.

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*}

Plug \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*} \begin{align*}D > 0\end{align*}, so there are **two real solutions.**

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

Plug \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*} \begin{align*}D = 0\end{align*}, so there is **one real solution.**

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

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

Plug \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*} \begin{align*}D < 0\end{align*}, so there are **no real solutions.**

**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*}, two non-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 tells us how many times its graph crosses the \begin{align*}x-\end{align*}axis. If \begin{align*}D > 0\end{align*}, the graph crosses the \begin{align*}x-\end{align*}axis in two places; if \begin{align*}D = 0\end{align*} it crosses in one place; if \begin{align*}D < 0\end{align*} it doesn’t cross at all:

#### Determining the Nature of Solutions

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

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

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

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

Re-write the equation in standard form: \begin{align*}10x^2-3x+4=0\end{align*}

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

Plug \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 0, 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 not only real, but also rational. If the discriminant is positive but not a perfect square, then the solutions to the equation are real but irrational.

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

Use the discriminant to determine the nature of the solutions.

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

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

Plug \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 positive but not a perfect square, so the solutions are **two real irrational numbers.**

The solutions to the equation are: \begin{align*}\frac{1 \pm \sqrt{21}}{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’ve seen 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.

#### Real-World Application: Football

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*}. If the goalpost is 10 feet high, can Marcus kick the ball high enough to go over the goalpost? What is the farthest distance that Marcus can kick the ball from and still make it over the goalpost?

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

Let \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.

**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 goalpost.

To find the value of \begin{align*}x\end{align*} that will work, we need to use the quadratic formula:

\begin{align*}x=\frac{-1 \pm \sqrt{0.8}}{-0.01}=189.4 \ feet \ \text{or} \ 10.56 \ feet\end{align*}

What does this answer mean? It means that if Marcus is exactly 189.4 feet or exactly 10.56 feet from the goalposts, the ball will just barely go over them. Are these the only distances that will work? No; those are just the distances at which the ball will be exactly 10 feet high, but *between* those two distances, the ball will go even higher than that. (It travels in a downward-opening parabola from the place where it is kicked to the spot where it hits the ground.) This means that Marcus will make the goal if he is anywhere **between 10.56 and 189.4 feet from the goalposts.**

### Example

#### Example 1

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*}, where \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?

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*}

Write the equation in standard form: \begin{align*}-0.003x^2+12x-12240=0\end{align*}

Find the discriminant: \begin{align*}D=(12)^2-4(-0.003)(-12240)=-2.88\end{align*}

Since the discriminant is negative, we know that **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

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*}
- \begin{align*}x^2+4x=2\end{align*}
- \begin{align*}-3x^2+2x+5=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*}
- \begin{align*}3x^2=7\end{align*}
- \begin{align*}x^2+30+225=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*}
- \begin{align*}2x^2-5x=12\end{align*}
- Marty is outside his apartment building. He needs to give his roommate 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 \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)

To view the Review answers, open this PDF file and look for section 10.10.