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What You Should Learn • Solve quadratic equations by factoring. • Solve quadratic equations by extracting square roots. • Solve quadratic equations by completing the square. • Use the Quadratic Formula to solve quadratic equations. • Use quadratic equations to model and solve real-life problems.

Factoring

Factoring A quadratic equation in x is an equation that can be written in the general form ax2 + bx + c = 0 where a, b, and c are real numbers with a  0. A quadratic equation in x is also called a second-degree polynomial equation in x. In this section, you will study four methods for solving quadratic equations: factoring, extracting square roots, completing the square, and the Quadratic Formula.

Factoring The first method is based on the Zero-Factor Property. If ab = 0, then a = 0 or b = 0. To use this property, write the left side of the general form of a quadratic equation as the product of two linear factors. Then find the solutions of the quadratic equation by setting each linear factor equal to zero. Zero-Factor Property

Example 1(a) – Solving a Quadratic Equation by Factoring 2x2 + 9x + 7 = 3 2x2 + 9x + 4 = 0 (2x + 1)(x + 4) = 0 2x + 1 = 0 x = x + 4 = 0 x = –4 The solutions are x = and x = –4. Check these in the original equation. Original equation Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0.

Example 1(b) – Solving a Quadratic Equation by Factoring cont’d 6x2 – 3x = 0 3x(2x – 1) = 0 3x = 0 x = 0 2x – 1 = 0 x = The solutions are x = 0 and x = . Check these in the original equation. Original equation Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0.

Factoring Be sure you see that the Zero-Factor Property works only for equations written in general form (in which the right side of the equation is zero). So, all terms must be collected on one side before factoring. For instance, in the equation (x – 5)(x + 2) = 8, it is incorrect to set each factor equal to 8. To solve this equation, you must multiply the binomials on the left side of the equation, and then subtract 8 from each side.

Factoring After simplifying the left side of the equation, you can use the Zero-Factor Property to solve the equation. Try to solve this equation correctly.

Extracting Square Roots

Extracting Square Roots Solving an equation of the form u2 = d without going through the steps of factoring is called extracting square roots.

Example 2 – Extracting Square Roots Solve each equation by extracting square roots. a. 4x2 = 12 b. (x – 3)2 = 7 Solution: a. 4x2 = 12 x2 = 3 x =  When you take the square root of a variable expression, you must account for both positive and negative solutions. Write original equation. Divide each side by 4. Extract square roots.

Example 2 – Solution cont’d So, the solutions are x = and x = – . Check these in the original equation. b. (x – 3)2 = 7 x – 3 =  x = 3  The solutions are x = 3  . Check these in the original equation. Write original equation. Extract square roots. Add 3 to each side.

Completing the Square

Completing the Square Note that when you complete the square to solve a quadratic equation, you are just rewriting the equation so it can be solved by extracting square roots.

Example 3 – Completing the Square: Leading Coefficient Is 1 Solve x2 + 2x – 6 = 0 by completing the square. Solution: x2 + 2x – 6 = 0 x2 + 2x = 6 x2 + 2x + 12 = 6 + 12 Write original equation. Add 6 to each side. Add 12 to each side.

Example 3 – Solution (x + 1)2 = 7 x + 1 =  x = –1  cont’d (x + 1)2 = 7 x + 1 =  x = –1  The solutions are x = –1  . Check these in the original equation as follows. Simplify. Take square root of each side. Subtract 1 from each side.

Example 3 – Solution Check: x2 + 2x – 6 = 0 cont’d Check: x2 + 2x – 6 = 0 (–1 + )2 + 2 (–1 + ) – 6 ≟ 0 8 – 2 – 2 + 2 – 6 ≟ 0 8 – 2 – 6 = 0 Check the second solution in the original equation. Write original equation. Substitute –1 + for x. Multiply. Solution checks.

Completing the Square When solving quadratic equations by completing the square, you must add (b/2)2 to each side in order to maintain equality. If the leading coefficient is not 1, you must divide each side of the equation by the leading coefficient before completing the square.

The Quadratic Formula

The Quadratic Formula The Quadratic Formula is one of the most important formulas in algebra. You should learn the verbal statement of the Quadratic Formula: “Negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.”

The Quadratic Formula In the Quadratic Formula, the quantity under the radical sign, b2 – 4ac, is called the discriminant of the quadratic expression ax2 + bx + c. It can be used to determine the nature of the solutions of a quadratic equation.

The Quadratic Formula If the discriminant of a quadratic equation is negative, as in case 3 above, then its square root is imaginary (not a real number) and the Quadratic Formula yields two complex solutions. When using the Quadratic Formula, remember that before the formula can be applied, you must first write the quadratic equation in general form.

Example 6 – The Quadratic Formula: Two Distinct Solutions Use the Quadratic Formula to solve x2 + 3x = 9. Solution: The general form of the equation is x2 + 3x – 9 = 0. The discriminant is b2 – 4ac = 9 + 36 = 45, which is positive. So, the equation has two real solutions. You can solve the equation as follows. x2 + 3x – 9 = 0 Write in general form. Quadratic Formula

Example 6 – Solution The two solutions are: cont’d The two solutions are: Check these in the original equation. Substitute a = 1, b = 3, and c = –9. Simplify. Simplify.

The Quadratic Formula: One Solution Use the Quadratic Formula to solve 8x2 – 24x + 18 = 0.