How to Solve Quadratic Equations in Vertex Form

By Petra Wakefield

The vertex form of a quadratic equation makes graphing the function straightforward. The vertex of the parabola is (h,k) from the vertex equation: y = a(x - h)^2 + k, where a, h and k are numerical constants. You need the vertex along with at least two other points to graph the parabola. One way to find the other points is by solving the equation for the x intercepts. This gives the same results you would get from applying the quadratic formula to the standard form of the equation.

Set y equal to zero in the equation. If your equation is y = 2(x + 3)^2 - 8, for example, it becomes 0 = 2(x + 3)^2 - 8 when y = 0.

Subtract k from both sides of the equation. For 0 = 2(x + 3)^2 - 8, k = -8, so subtract -8 from both sides. Remember subtracting a negative is the same as adding a positive, so 0 - (-8) = 2(x + 3)^2 - 8 - (-8) becomes 8 = 2(x + 3)^2.

If the number on the left side of the equation is negative, the equation does not have any solutions. Stop here and record the result as "no solution."

Divide both sides of the equation by a to isolate the term containing the variable. For 8 = 2(x + 3)^2, a = 2, so divide both sides by 2 to get 4 = (x + 3)^2.

Take the square root of both sides of the equation. Remember that the square root of a positive number can be positive or negative, so for 4 = (x + 3)^2, you get x + 3 = 2 or -2.

Subtract the constant on the left side of the equation from both sides to find the values of x. For x + 3 = 2 or -2, subtracting 3 from both sides gives x + 3 - 3 = 2 - 3 or -2 - 3, so x = -1 and x = -5. Record the solution with the lower value first; in this case, the solution is x = -5, -1.

About the Author

Petra Wakefield is a writing professional whose work appears on various websites, focusing primarily on topics about science, fitness and outdoor activities. She holds a Master of Science in agricultural engineering from Texas A&M University.