Deriving the Quadratic Formula
The “horrible looking” quadratic formula below
is actually derived using the steps involved in completing the square. It stems from the fact that any generic quadratic function of the form y = ax2 + bx + c can be solved by finding its roots, or the points at which the graph of the parabola hits the x-axis known as the x-intercepts.
So to find the roots or x-intercepts of y = ax2 + bx + c, we need to let y = 0. That means we have
From here, I am going to apply the usual steps involved in completing the square to arrive at the quadratic formula.
Derivation of the quadratic formula is easy! Here we go…
- Step 1: Let y = 0 in the generic quadratic function y = ax2 + bx + c
- Step 2: Move the constant “c” to the right side of the equation by subtracting both sides by “c“.
- Step 3: Divide the entire equation by the coefficient of the squared term which is “a“.
- Step 4: Now identify the coefficient of the linear term x.
- Step 5: Divide it by 2 and raise it to the 2nd power.
- Step 6: Add the output of step #5 to both sides of the equation.
- Step 7: Simplify the right side of the equation. Be careful when you add fractions. Make sure that you find the LCD.
- Step 8: Express the trinomial on the left side of the equation as the square of a binomial.
- Step 9: Apply square root operations in both sides of the equation to eliminate the exponent 2 of the binomial.
- Step 10: Simplify. The left side no longer contains the power 2.
- Step 11: Keep the variable “x” on the left side by subtracting both sides by .
- Step 12: Simplify and we are done!
I hope that you find the step by step solution helpful in figuring out how the quadratic formula is derived using the method of completing the square.