Suppose that the balance of your checking account in dollars can be modeled by the function \begin{align*}B(t)=0.001t^2 - t + 300\end{align*}, where \begin{align*}t\end{align*} is the number of days the checking account has been open. Will the balance of your checking account ever be $100?

### Solutions Using the Discriminant

You have seen parabolas that intersect the \begin{align*}x-\end{align*}axis twice, once, or not at all. There is a relationship between the number of real \begin{align*}x-\end{align*}intercepts and the quadratic formula.

**Case 1:** The parabola has two \begin{align*}x-\end{align*}intercepts. This situation has two possible solutions for \begin{align*}x\end{align*}, because the value inside the square root is positive. Using the quadratic formula, the solutions are \begin{align*}x=\frac{-b+\sqrt{b^2-4ac}}{2a}\end{align*} and \begin{align*}x=\frac{-b-\sqrt{b^2-4ac}}{2a}\end{align*}.

**Case 2:** The parabola has one \begin{align*}x-\end{align*}intercept. This situation occurs when the vertex of the parabola just touches the \begin{align*}x-\end{align*}axis. This is called a repeated root, or **double root.** The value inside the square root is zero. Using the quadratic formula, the solution is \begin{align*}x=\frac{-b}{2a}\end{align*}.

**Case 3:** The parabola has no \begin{align*}x-\end{align*}intercept. This situation occurs when the parabola does not cross the \begin{align*}x-\end{align*}axis. The value inside the square root is negative, so there are no real roots. The solutions to this type of situation are **imaginary,** which you will learn more about in a later textbook.

The value inside the square root of the quadratic formula is called the **discriminant.** It is symbolized by \begin{align*}D\end{align*}. It dictates the number of real solutions the quadratic equation has. This can be summarized with the Discriminant Theorem.

- If \begin{align*}D>0\end{align*}, the parabola will have two \begin{align*}x-\end{align*}intercepts. The quadratic equation will have two real solutions.
- If \begin{align*}D=0\end{align*}, the parabola will have one \begin{align*}x-\end{align*}intercept. The quadratic equation will have one real solution.
- If \begin{align*}D<0\end{align*}, the parabola will have no \begin{align*}x-\end{align*}intercepts. The quadratic equation will have zero real solutions.

#### Let's determine the number of solutions for the following equations:

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

By finding the value of its discriminant, you can determine the number of \begin{align*}x-\end{align*}intercepts the parabola has and thus the number of real solutions.

\begin{align*}D &= b^2-4(a)(c)\\ D &= (4)^2-4(-3)(1)\\ D &= 16+12=28\end{align*}

Because the discriminant is positive, the parabola has two real \begin{align*}x-\end{align*}intercepts and thus two real solutions.

- Determine the number of solutions to \begin{align*}-2x^2+x=4\end{align*}.

Before we can find its discriminant, we must write the equation in standard form: \begin{align*}ax^2+bx+c=0\end{align*}.

Subtract 4 from each side of the equation: \begin{align*}-2x^2+x-4=0\end{align*}.

\begin{align*}\text{Find the discriminant:} && D &= (1)^2-4(-2)(-4)\\ && D &= 1-32=-31\end{align*}

The value of the discriminant is negative; there are no real solutions to this quadratic equation. The parabola does not cross the \begin{align*}x-\end{align*}axis.

#### Now, let's use the discriminant to solve the following problem:

Emma and Brandon 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+27,760\end{align*}, where \begin{align*}x\end{align*} represents the number of helmets produced. Their goal is to make a profit of $40,000 this year. Is this possible?

The equation we are using is \begin{align*}40,000=0.003x^2+12x+27,760\end{align*}. By finding the value of its discriminant, you can determine if the profit is possible.

Begin by writing this equation in standard form:

\begin{align*}0 &= 0.003x^2+12x-12,240\\ D &= b^2-4(a)(c)\\ D &= (12)^2-4(0.003)(-12,240)\\ D &= 144+146.88=290.88\end{align*}

Because the discriminant is positive, the parabola has two real solutions. Yes, the profit of $40,000 is possible.

### Examples

#### Example 1

Earlier, you were told that the balance of your checking account in dollars can be modeled by the function \begin{align*}B(t)=0.001t^2 - t + 300\end{align*}, where \begin{align*}t\end{align*} is the number of days the checking account has been open. Will the balance of your checking account ever be $100?

You can answer this problem by finding the discriminant.

\begin{align*}D &= b^2-4(a)(c)\\ D &= (-1)^2-4(.001)(300)\\ D &= 1-1.2=-.2\end{align*}

The discriminant is negative, so the function has 0 real solutions. Therefore, the balance of your checking account will never be $100.

#### Example 2

Determine the number of solutions for \begin{align*}x^2-2x+1=0\end{align*}.

Substitute the values into the discriminant:

\begin{align*}D &= b^2-4(a)(c)\\ D &= (-2)^2-4(1)(1)\\ D &= 4-4=0\end{align*}

Because the discriminant is zero, the parabola has one real \begin{align*}x-\end{align*}intercept and thus one real solution.

### Review

- What is a discriminant? What does it do?
- What is the formula for the discriminant?
- Can you find the discriminant of a linear equation? Explain your reasoning.
- Suppose \begin{align*}D=0\end{align*}. Draw a sketch of this graph and determine the number of real solutions.
- \begin{align*}D=-2.85\end{align*}. Draw a possible sketch of this parabola. What is the number of real solutions to this quadratic equation.
- \begin{align*}D>0\end{align*}. Draw a sketch of this parabola and determine the number of real solutions.

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*}

A solution to a quadratic equation will be irrational if the discriminant is **not a perfect square.** If the discriminant is a perfect square, then the solutions will be rational numbers. Using the discriminant, 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?
- 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 Marcus kicked the ball hard enough to go over the goal post, which is 10 feet high.

**Mixed Review**

- Factor \begin{align*}6x^2-x-12\end{align*}.
- Find the vertex of \begin{align*}y=-\frac{1}{4} x^2-3x-12\end{align*} by completing the square.
- Solve using the quadratic formula: \begin{align*}-4x^2-15=-4x\end{align*}.
- How many centimeters are in four fathoms? (Hint: 1 fathom = 6 feet)
- Graph the solution to \begin{align*}\begin{cases} 3x+2y \le -4\\ x-y>-3 \end{cases}\end{align*}.
- How many ways can 3 toppings be chosen from 7 options?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.9.