How to Convert Quadratic Equations From Standard to Vertex Form

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The standard form of a quadratic equation is y = ax^2 + bx + c, where a, b, and c are coefficiencts and y and x are variables. It is easier to solve a quadratic equation when it is in standard form because you compute the solution with a, b, and c. However, if you need to graph a quadratic function, or parabola, the process is streamlined when the equation is in vertex form. The vertex form of a quadratic equation is y = m(x-h)^2 + k with m representing the slope of the line and h and k as any point on the line.

Factor Coefficient

Factor the coefficient a from the first two terms of the standard form equation and place it outside of the parentheses. Factoring standard form quadratic equations involves finding a pair of numbers that add up to b and multiply to ac. For instance, if you are converting 2x^2 - 28x + 10 to vertex form, you first need to write 2(x^2 - 14x) + 10.

Divide Coefficient

Next, divide the coefficient of the x term inside the parentheses by two. Use the square root property to then square that number. Using that square root property method helps to find the quadratic equation solution by taking the square roots of both sides. In the example, the coefficient of the x inside the parentheses is -14.

Balance Equation

Add the number inside the parentheses, and then to balance the equation, multiply it by the factor on the outside of parentheses and subtract this number from the whole quadratic equation. For example, 2(x^2 - 14x) + 10 becomes 2(x^2 - 14x + 49) + 10 - 98, since 49*2 = 98. Simplify the equation by combining the terms at the end. For example, 2(x^2 - 14x + 49) - 88, since 10 - 98 = -88.

Convert Terms

Finally, convert the terms inside parentheses to a squared unit of the form (x - h)^2. The value of h is equal to half the coefficient of the x term. For example, 2(x^2 - 14x + 49) - 88 becomes 2(x - 7)^2 - 88. The quadratic equation is now in vertex form. Graphing the parabola in vertex form requires the use of the symmetric properties of the function by first choosing a left side value and finding the y variable. You can then plot the data points to graph the parabola.


About the Author

Nucreisha Langdon has written professionally since 1991. She has ghostwritten more than 20 romantic fantasy novels, while her nonfiction work has appeared in the "Gainesville Sun" and the "Austin Chronicle." Langdon holds a Bachelor of Science in mathematics and a Bachelor of Arts in English from the University of Florida.