# Example of a Quadratic Equation with One Real Solution

This video explains how to solve a quadratic equation with one solution. The quadratic equation 4x^{2}+bx+9=0 has only one solution, meaning the discriminant must be equal to 0. Using this information, and knowing what a and b are, the solution can be found.

## Quadratic Equation with One Solution

The quadratic equation 4x squared, plus bx, plus 9, equals 0, has only one solution. Find the solution. There’s a problem here. They put a b instead of an actual number. We don’t know what b is. If we don’t know what b is, then we can’t find our solution. We can find b though, because they gave us a clue.

They told us that this quadratic equation only has one solution. That means that the discriminant, b squared minus 4ac, must be equal to 0. Using that information, and since we know a and c, we can find our b. b squared minus 4, times a, is the leading coefficient. Times 4, then times c, which is 9, equals 0. b squared minus 4, times 9, is 36, times 4, is 144, equals 0.

Then we add 144 to both sides to get b alone. Those cancel because they’re added inverses. b squared is equal to 144. To solve for b we must square root both sides. The square root of b squared is b, and the square root of 144 is 12. Now that we know b is, we can use the quadratic formula to find our solution.

That is x equals -b, plus or minus the square root of b squared minus 4ac, that’s the discriminant right there, all divided by 2a. Again, since we know it only has one solution, that meant that our discriminant was 0. This is just the square root of 0, and the square root of 0 is just 0.

Really we can get rid of this whole thing right here, and we just have that x equals -b, divided by 2a. x is the opposite of b, we found b earlier, b is 12. -12 divided by 2, times a, which is 4. That’s -12 divided by 8, which simplifies, we can divide the numerator and the denominator by 4.

-12 divided by 4, is -3, and 8 divided by 4 is 2. The solution is x equals -3/2. That’s where our parabola intersects the x axis in this case. This would be the x value of our Vertex. It’s also the solution of the quadratic equation.