D. Solving Quadratic Equations using the Quadratic Formula

Sometimes factoring a quadratic expression is difficult and many quadratic expressions cannot be factored using integers. Fortunately, the quadratic formula can be used to find the solutions to all quadratic equations, even those involving quadratic expressions that are not factorable.

quadratic formula:

In 628 AD, an Indian mathematician by the name of Brahmagupta derived the quadratic formula from the standard form of the quadratic equation, . To derive the quadratic formula, he isolated x algebraically.

Optional activity: Deriving the Quadratic Formula

Quadratic Formula Standard form for a
Quadratic Equation

When solving a quadratic equation written in standard form, simply enter the coefficients into the quadratic formula to determine the solution(s) to the equation.

Example 1

Solve the equation using the quadratic formula and express the roots as exact values.

Step 1 : Substitute the values of a , b , and c into the quadratic formula.


Step 2 : Simplify.

Step 3 : State each root individually, as exact values.

and

Step 4 : Verify by substituting the solutions into the original equation.

and

For ,

Left side
Right side
0
LS = RS

For ,

Left side
Right side
0
LS = RS

The solutions to the quadratic equation are and .

Please refer to pages 414–418, Examples 1 and 4, of Principles of Mathematics 11 for more examples using the quadratic formula.