# If the discriminant of a quadratic equation is equal to zero, how many real solutions does the...

## Question:

If the discriminant of a quadratic equation is equal to zero, how many real solutions does the equation have?

## Quadratic Equations: Discriminant

Let's say that we are interested in solving a quadratic equation described by {eq}a_1x^2 + a_2x + a_3 = 0 {/eq}. Such a quadratic equation in a standard form can be solved using the quadratic formula which involves a discriminant value in it. The discriminant for this equation is given by:

{eq}D = a_2^2 - 4a_1a_3 {/eq}

According to the quadratic formula, the two solutions then are {eq}x_{1, 2} = \dfrac{-a_2 \pm \sqrt{D}}{2a_1} {/eq}

## Answer and Explanation:

Let's say that our quadratic equation is:

{eq}ax^2 + bx + c = 0 {/eq}

According to the quadratic formula, the two solutions are:

{eq}x_{1, 2} = \dfrac{-b \pm \sqrt{D}}{2a} {/eq}

If the value of the discriminant is 0, then:

{eq}D = 0\\ \sqrt{D} = 0 {/eq}

Which reduces the formula to:

{eq}x_{1, 2} = \dfrac{-b}{2a} {/eq}

Notice that we still have two solutions, although the two solutions have exactly same value. We can conclude that if the value of the discriminant is 0, then we have **one real solution with a multiplicity of 2**.

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from Math 101: College Algebra

Chapter 4 / Lesson 10