Section 6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula

What You Will Learn Solving Quadratic Equations by Using Factoring Solving Quadratic Equations by Using the Quadratic Formula

Binomial A binomial is an expression that contains two terms in which each exponent that appears on a variable is a whole number. x + 3, x – 5, 3x + 5, 4x – 2

FOIL To multiply two binomials, we use the FOIL method. F = First O = Outer I = Inner L = Last

Example 2: Multiplying Binomials Multiply (2x + 4)(x + 6). Solution F O I L

Trinomial A trinomial is an expression containing three terms in which each exponent that appears on a variable is a whole number. x2 + 8x + 15, x2 + 11x + 28

To Factor Trinomial Expressions of the Form x2 + bx + c 1. Find two numbers whose product is c and whose sum is b. 2. Write the factors in the form 3. Check your answer by multiplying the factors using the FOIL method.

Example 4: Factoring A Trinomial Factor x2 – 6x – 16. We need to find two numbers whose product is –16 and whose sum is –6. Factors of 12 Sum of Factors –16(1) –16 + 1 = –15 –8(2) –8 + 2 = –6 –4(4) –4 + 4 = 0 –2(8) –2 + 8 = 6 –1(16) –1 + 16 = 15 x2 – 6x – 16 = (x – 8)(x + 2)

Factoring Trinomials of the Form ax2 + bc + c, a ≠ 1. 1. Write all pairs of factors of the coefficient of the squared term, a. 2. Write all pairs of the factors of the constant, c. 3. Try various combinations of these factors until the sum of the products of the outer and inner terms is bx. 4. Check your answer by multiplying the factors using the FOIL method.

Example 6: Factoring A Trinomial, a ≠ 1 Factor 6x2 – 11x – 10. Solution The factors will be either 6 • 1 or 2 • 3. Try the medium sized ones first. 6x2 – 11x – 10 = (2x )(3x ) –10: (–1)(10), (1)(10), (–2)(5), (2)(–5) Try the eight pairs to find 6x2 – 11x – 10 = (2x – 5)(3x + 2)

Solving Quadratic Equations by Factoring Standard Form of a Quadratic Equation ax2 + bx + c = 0, a ≠ 0

Solving Quadratic Equations by Factoring Zero-Factor Property If a • b = 0, then a = 0 or b = 0.

To Solve a Quadratic Equation by Factoring 1. Use the addition or subtraction property to make one side of the equation equal to 0. 2. Factor the side of the equation not equal to 0. 3. Use the zero-factor property to solve the equation.

Example 9: Solving a Quadratic Equation by Factoring Solve the equation 3x2 – 7x – 6 = 0. Solution The solutions are –2/3 and 3.

Solving Quadratic Equations by Using the Quadratic Formula When a quadratic equation cannot be easily solved by factoring, we can solve the equation with the quadratic formula. The quadratic formula can be used to solve any quadratic equation.

Quadratic Formula For a quadratic equation in standard form, ax2 + bx + c = 0, a ≠ 0, the quadratic formula is

No Real Solution It is possible for a quadratic equation to have no real solution. In solving an equation, if the radicand (the expression inside the square root) is a negative number, the quadratic equation has no real solution.

To Solve a Quadratic Equation by Using the Quadratic Formula 1. Write the equation in standard form. 2. Determine the values for a (the coefficient of the squared term), b (the coefficient of the x term), and c (the constant). 3. Substitute the values of a, b, and c into the quadratic formula and evaluate the expression.

Example 11: Irrational Solutions to a Quadratic Equation Solve 4x2 – 8x = –1 by using the quadratic formula. Solution Write the equation in standard form: 4x2 – 8x + 1 = 0 Identify a, b, and c: a = 4, b = –8, c = 1 Substitute into the quadratic formula:

Example 11: Irrational Solutions to a Quadratic Equation a = 4, b = –8, c = 1

Example 11: Irrational Solutions to a Quadratic Equation The solutions are and