You can use the quadratic formula to solve any quadratic equation. A quadratic equation can have two distinct real number roots or one distinct real number root. These roots represent where the graph of a quadratic equation crosses or touches the x-axis.
A quadratic equation may also have two complex number roots. A complex number solution indicates that the graph of the quadratic equation does not cross or touch the x-axis. Remember that a real number can be graphed on a number line. A complex number has the form \( a + bi \) where \( i = \sqrt{- 1}. \)
You can use the discriminant to determine the number and type of roots a quadratic equation will have before solving it. The discriminant is the value of the expression under the radical in the quadratic formula.
Let's Watch
In the video below, the instructor will detail information about the discriminant and how it is used in solving quadratic equations. Pay close attention to how he determines the discriminant and then solves the quadratic equations.
You may want to follow along using the study guide from earlier in the lesson.
The computation inside the radical is known as the discriminant. This is sometimes determined first because the value of the discriminant helps us to understand the nature of the solutions to the quadratic equation. When the discriminant is positive, there will be two real-number solutions. When the discriminant is zero, there will be one real-number solution. When the discriminant is negative, there will be no real-number solutions, but rather, two complex-number solutions. But why are these statements true? If you think about it, one must find the square root of the discriminant in the quadratic formula. And the square root of a positive number has two real-number solutions. The square root of zero has one solution, and that’s zero. Where as the square root of a negative number has no real-number solutions, but two imaginary-number solutions.
We can consider the quadratic formula written this way when computing the discriminant first. In these examples, I would like to determine the discriminant and nature of the roots first, then solve for X. I like to write down the values for A, B, and C, then compute D. (silent). Since the discriminant is positive, I know we will have two real-number solutions. We can now identify them very quickly (silent), and if needed determine the decimal approximations. Please pause the video now and try the last two examples on your own. Resume playback in a moment to check your work. Good Luck!
Martyn is solving a quadratic equation in which the discriminant is negative. What does this mean about the number and type of solutions of the quadratic equation?
Ceri says that when a quadratic equation has only 1 root, the graph of that quadratic equation touches the x-axis at only one point. Is Ceri correct? Explain.
The discriminant of a certain quadratic equation is greater than 0. What is true about the graph of that quadratic equation?
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When \( d < 0, \) the quadratic equation will have two complex number solutions. |
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The solution(s) of a quadratic equation is the x-value at the point where the graph of the equation crosses or touches the x-axis. If a quadratic equation has only one solution, then that graph touches the x-axis at only one point (the vertex). |
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When \( d > 0, \) the quadratic equation will have two real number solutions. The graph of the quadratic equation will cross the x-axis at each of the solutions. |
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