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Solving Quadratic Equations by Factoring 6.6 Solving Quadratic Equations by Factoring

Quadratic Equation Quadratic Equation A quadratic equation is one that can be written in the form ax2 + bx + c = 0, where a, b and c are real numbers and a ≠ 0. The form ax2 + bx + c = 0 is referred to as standard form.

Zero-Factor Property Zero-Factor Property If a and b are real numbers and if ab = 0, then a = 0 or b = 0.

Example Solve: (x – 4)(x + 2) = 0 x – 4 = 0 or x + 2 = 0 You can check the solutions on your own.

Solving Quadratic Equations To Solve Quadratic Equations by Factoring Step 1: Write the equation in standard form. Step 2: Factor the quadratic completely. Step 3: Set each factor containing a variable equal to 0. Step 4: Solve the resulting equations. Step 5: Check each solution in the original equation.

Example Solve: x2 – 5x = 24 First write the quadratic equation in standard form. x2 – 5x – 24 = 0 Now factor. x2 – 5x – 24 = (x – 8)(x + 3) = 0 Set each factor equal to 0. x – 8 = 0 or x + 3 = 0, which will simplify to x = 8 or x = –3 Continued

Example (cont) Check both possible answers in the original equation. 82 – 5(8) = 64 – 40 = 24 True (–3)2 – 5(–3) = 9 – (–15) = 24 True So the solutions are 8 or –3. ?

Example Solve: 4x(8x + 9) = 5 4x(8x + 9) = 5 32x2 + 36x = 5 8x – 1 = 0 or 4x + 5 = 0 8x = 1 4x = – 5 x = x = The solutions are and Continued

Example (cont) Check both possible answers in the original equation. True ? True ? The solutions are and

Example Solve: Replace x with each solution in the original equation. The solutions all check.

Finding x-intercepts Recall that in Chapter 3, we found the x-intercept of linear equations by letting y = 0 and solving for x. The same method works for x-intercepts in quadratic equations. Note: When the quadratic equation is written in standard form, the graph is a parabola opening up (when a > 0) or down (when a < 0), where a is the coefficient of the x2 term. The intercepts will be where the parabola crosses the x-axis.

Example Find the x-intercepts of the graph of y = 4x2 + 11x + 6. The equation is already written in standard form, so we let y = 0, then factor the quadratic in x. 0 = 4x2 + 11x + 6 = (4x + 3)(x + 2) We set each factor equal to 0 and solve for x. 4x + 3 = 0 or x + 2 = 0 4x = –3 or x = –2 x = –¾ or x = –2 So the x-intercepts are the points (–¾, 0) and (–2, 0).