*9.1 Completing the Square and the Quadratic

Formula

*Solve.

1. 2. x2 x 6 0 3x2 11x 4 0

*Often times we are not able to a quadratic equation in

order to solve it. When this is the case, we have two other methods: completing the square and the quadratic formula.

*By learning how to the we can force a quadratic

expression to factor.

factor

completesquare

*Steps for Solving a Quadratic by Completing the Square

*1. Add or subtract the constant term to the other side (if necessary).

*2. Check to make sure the coefficient of is . If not, factor out the coefficient of and divide both sides of the equation by this number.

*3. Take of b, square it, and add it to sides.

*4. Make the left side a square of a binomial (example: ).

*5. Simplify the right side.

*6. Take the square root of each side. (make sure to use ).

*7. Solve for x.

x2

x2

x 1 2

1

half both

±

*Solve by Completing the Square

*3. x2 2x 13

*Solve by Completing the Square

*4. x2 10x 16 0

*Solve by Completing the Square

*5.p2

2

5p

1

5

*Solve by Completing the Square

*6. 3e2 16e 8 0

*The Quadratic Formula*The solutions of a quadratic equation in general form

, when , are given by the

quadratic formula:

ax2 bx c 0 a 0

x b b2 4ac

2a

*Steps to Solving the Quadratic Formula

*1. Write the equation in the form

*2. Determine the values of a, b, and c.

*3. Substitute the values of a, b, and c. into the quadratic formula and evaluate the expression.

*4. The sign indicates that there are two solutions of the equation.

ax2 bx c 0

*Solve using the Quadratic Equation

*7. 2x2 6x 10

*Solve using the Quadratic Equation

*8. 3x2 2x 4 0

*Solve using the Quadratic Equation

*9. 4x2 2x 7

*Solve using the Quadratic Equation

*10. 3x2 60