y completing the square of the quadratic equation ax2 + bx + c = 0, where a ≠ 0, a formula can be developed that gives the solutions to any quadratic equation in standard form. The formula is called Quadratic Formula.
Step Illustrative Example 1. Place the constant term on the right
side of the equation. All the terms with 2 + + = 0 unknowns are on the left side.
2. The numerical coefficient of x2 should be 1. Divide each term of the equation with the numerical coefficient of x2 if necessary.
2 + = −
2 + = −
3. To get the constant term needed to complete the square, get the numerical coefficient of x, divide it by 2 and square it. Add the result to both sides of the equation.
2 2 2 ++42 = +42
4. Factor the perfect square trinomial.
6. Equate the linear expressions to each of the two values.
( + )2 = 2−4
2
+ 2
42
= ± √2−4 2
5. Extract the square root from both sides. Two values will be obtained for the right side of the equation.
+ =±√2−4 2 42
+ =±√2−4 2 √42
7. Solve each of the resulting linear equations.
= − ± √2 − 4 , ≠ 0
2
This is the quadratic formula.
Example 1: Solve for the real roots of each equation using quadratic formula. a. 2 – 2 – 19 = 0
b. 22 + 6 = −3
c. 22–3+4=0
Solutions
a. In2–2–19 = 0, = 1, = −2,and = −19,
= −±√2−4
2
= −(−2)±√(−2)2−4(1)(−19)
Quadratic formula
Substitute the values of a, b and c Simplify
Evaluate the square root, if possible.