**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 = ax^{2} + 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* = *ax*^{2} + *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*=*ax*^{2}+*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.