How to Use the Quadratic Formula to Find Roots of Equations

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  • 0:03 Quadratic Equation and Roots
  • 1:36 Quadratic Formula
  • 3:28 Another Example
  • 4:30 Lesson Summary
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Lesson Transcript
Instructor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Expert Contributor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we'll look at quadratic equations and learn how to find the roots of these equations using the quadratic formula. We'll use the definition and some example to become comfortable using this invaluable tool.

Quadratic Equations and Roots

Interesting fact: when a ball is thrown in the air, its trajectory can be modeled by a quadratic equation. This means that you can use the equation to determine different things about the path the ball takes. Not only that, but you can determine how long it will take for the ball to hit the ground!

A quadratic equation is an equation where the highest exponent of any variable is 2:


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Most of the time, we write a quadratic equation in the form ax2 + bx + c = 0, and the values of x that make the equation true are called roots of the equation. Quadratic equations have 2 roots; you can remember this because the highest exponent of x is also 2!

Back to our interesting fact. Suppose we are standing on the top of a podium that is 5 feet tall, and we throw a ball in the air.


quaform6


Its trajectory can be modeled using the equation y = -x2 + 4x + 5, where y represents the ball's height above the ground at x seconds. Any ideas on how we could find out when the ball hits the ground below? In other words, how long is the ball in the air?

Let's think about it. Since y represents the ball's height above the ground, it should be the case that when the ball hits the ground, y would be 0. Ah ha! I've got an idea! Let's plug zero into our equation for y, since that 's when the ball will hit the ground:

-x2 + 4x + 5 = 0

Great! Now, if we can just figure out what values of x make this true or, in other words, if we can find the roots of the equation, then we will know at what time the ball will hit the ground. Let's look at a way to find these roots.

The Quadratic Formula

The quadratic formula is a formula that we can use to find the roots of the quadratic equation ax2 + bx + c = 0.


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To use the quadratic formula to find the roots of a quadratic equation, all we have to do is get our quadratic equation into the form ax2 + bx + c = 0; identify a, b, and c; and then plug them in to the formula. To identify these values, we just remember that a is in front of x2, b is in front of x, and c is the number by itself.

For example, in our equation -x2 + 4x + 5 = 0, the number in front of x2 is -1, so a = -1. The number in front of x is 4, so b = 4. Lastly, the number by itself is 5, so c = 5. We're almost there! All we have to do is plug these values into our quadratic formula, and then we can find the values of x that make our equation true. Then we will know how long it takes for the ball to hit the ground. Let's get to plugging!


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Additional Activities

Exploring the Discriminant of the Quadratic Formula and What it Tells About the Roots of a Quadratic Equation

Notes:

  • The discriminant of the quadratic formula is the portion of the formula that is under the square root, given by the formula b2 - 4ac.
  • There are three possibilities for the value of the discriminant of the quadratic formula, and those are a positive value, a negative value, and a value of 0.


Questions:

  1. Calculate the determinant of x2 + 2x + 1 = 0. Determine if the discriminant is positive, negative, or 0. Use the quadratic formula to find the roots of this equation, and determine how many real roots the equation has.
  2. Calculate the determinant of 2x2 - 4x + 3 = 0. Determine if the discriminant is positive, negative, or 0. Use the quadratic formula to find the roots of this equation, and determine how many real roots the equation has.
  3. Calculate the determinant of x2 - 6x + 8 = 0. Determine if the discriminant is positive, negative, or 0. Use the quadratic formula to find the roots of this equation, and determine how many real roots the equation has.
  4. Based on your findings in questions 1, 2, and 3, what theories could you come up with regarding the value of the discriminant of a quadratic equation and the number of real roots that the quadratic equation has.
  5. Test this theory on the following quadratic equations: 4x2 + 4x + 1 = 0, x2 + 7x - 30 = 0, and 3x2 + 2x + 9 = 0.


Solutions:

  • Solution to problem 1: The determinant of x2 + 2x + 1 = 0 is 0. The quadratic formula gives that the root of this equation is -1, so it has just one real root.
  • Solution to problem 2: The determinant of 2x2 - 4x + 3 = 0 is -8, so it has a negative value. The quadratic formula gives that the roots of this equation not real numbers, so this equation has no real roots.
  • Solution to problem 3: The determinant of x2 - 6x + 8 = 0 is 4, so it has a positive value. The quadratic formula gives that the roots of this equation are 2 and 4, and both of these are real, so the equation has two real roots.
  • Solution to problem 4: If the discriminant is positive, then the equation has two real roots. If the discriminant is negative, then the equation has no real roots. If the discriminant is 0, then the equation has exactly one real root.
  • Solution to problem 5: The discriminant of 4x2 + 4x + 1 = 0 is 0, and the quadratic formula gives exactly one real root of {eq}-\frac{1}{2} {/eq}. The discriminant of x2 + 7x - 30 = 0 is 169, and the quadratic formula gives two real roots of -10 and 3. The discriminant of 3x2 + 2x + 9 = 0 is -104, and the quadratic formula gives two roots that are not real. Thus, the rules hold up in all of these cases.

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