10.9: Solutions Using the Discriminant
Suppose that the balance of your checking account in dollars can be modeled by the function
Watch This
Multimedia Link: This http://sciencestage.com/v/20592/alevelmaths:rootsofaquadraticequation:discriminant:examsolutions.html  video, presented by Science Stage, helps further explain the discriminant using the Quadratic Formula.
Guidance
You have seen parabolas that intersect the
Case 1: The parabola has two
Case 2: The parabola has one
Case 3: The parabola has no
The value inside the square root of the quadratic formula is called the discriminant. It is symbolized by
 If
D>0 , the parabola will have twox− intercepts. The quadratic equation will have two real solutions.  If
D=0 , the parabola will have onex− intercept. The quadratic equation will have one real solution.  If
D<0 , the parabola will have nox− intercepts. The quadratic equation will have zero real solutions.
Example A
Determine the number of real solutions to
Solution: By finding the value of its discriminant, you can determine the number of
Because the discriminant is positive, the parabola has two real
Example B
Determine the number of solutions to
Solution: Before we can find its discriminant, we must write the equation in standard form:
Subtract 4 from each side of the equation:
The value of the discriminant is negative; there are no real solutions to this quadratic equation. The parabola does not cross the
Example C
Emma and Bradon own a factory that produces bike helmets. Their accountant says that their profit per year is given by the function
Solution: The equation we are using is
Begin by writing this equation in standard form:
Because the discriminant is positive, the parabola has two real solutions. Yes, the profit of $40,000 is possible.
Guided Practice
Determine the number of solutions for
Solution:
Substitute the values into the discriminant:
Because the discriminant is zero, the parabola has one real
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Discriminant of Quadratic Equations (10:14)
 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
D=0 . Draw a sketch of this graph and determine the number of real solutions. 
D=−2.85 . Draw a possible sketch of this parabola. What is the number of real solutions to this quadratic equation. 
D>0 . Draw a sketch of this parabola and determine the number of real solutions.
Find the discriminant of each quadratic equation.

2x2−4x+5=0 
x2−5x=8 
4x2−12x+9=0 
x2+3x+2=0 
x2−16x=32 
−5x2+5x−6=0
Determine the nature of the solutions of each quadratic equation.

−x2+3x−6=0 
5x2=6x 
41x2−31x−52=0 
x2−8x+16=0 
−x2+3x−10=0  \begin{align*}x^264=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+2x3=0\end{align*}
 \begin{align*}3x^211x=10\end{align*}
 \begin{align*}\frac{1}{2}x^2+2x+\frac{2}{3}=0\end{align*}
 \begin{align*}x^210x+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(2000.4x)\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^2x12\end{align*}.
 Find the vertex of \begin{align*}y=\frac{1}{4} x^23x12\end{align*} by completing the square.
 Solve using the quadratic formula: \begin{align*}4x^215=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\\ xy>3 \end{cases}\end{align*}.
 How many ways can 3 toppings be chosen from 7 options?
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discriminant
The discriminant is the part of the quadratic formula under the radical, . A positive discriminant suggests two real roots to the quadratic equation, a zero suggests one real root with multiplicity two, and a negative indicates two complex roots.Double Root
A solution that is repeated twice.Image Attributions
Here you'll learn how to find the number of solutions to a quadratic equation by calculating the discriminant and how to use this information to solve problems.