# 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?

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}

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.