2 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.
4 FactoringA quadratic equation in x is an equation that can be written in the general formax2 + bx + c = 0where 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.
5 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
6 Example 1(a) – Solving a Quadratic Equation by Factoring 2x2 + 9x + 7 = 32x2 + 9x + 4 = 0(2x + 1)(x + 4) = 02x + 1 = x =x + 4 = x = –4The solutions are x = and x = –4. Check these in the original equation.Original equationWrite in general form.Factor.Set 1st factor equal to 0.Set 2nd factor equal to 0.
7 Example 1(b) – Solving a Quadratic Equation by Factoring cont’d6x2 – 3x = 03x(2x – 1) = 03x = x = 02x – 1 = x =The solutions are x = 0 and x = . Check these in the original equation.Original equationFactor.Set 1st factor equal to 0.Set 2nd factor equal to 0.
8 FactoringBe 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.
9 FactoringAfter simplifying the left side of the equation, you can use the Zero-Factor Property to solve the equation.Try to solve this equation correctly.
11 Extracting Square Roots Solving an equation of the form u2 = d without going through the steps of factoring is called extracting square roots.
12 Example 2 – Extracting Square Roots Solve each equation by extracting square roots.a. 4x2 = b. (x – 3)2 = 7Solution:a. 4x2 = 12x2 = 3x = 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.
13 Example 2 – Solutioncont’dSo, the solutions are x = and x = – Check these in the original equation.b. (x – 3)2 = 7x – 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.
15 Completing the SquareNote 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.
16 Example 3 – Completing the Square: Leading Coefficient Is 1 Solve x2 + 2x – 6 = 0 by completing the square.Solution:x2 + 2x – 6 = 0x2 + 2x = 6x2 + 2x + 12 =Write original equation.Add 6 to each side.Add 12 to each side.
17 Example 3 – Solution (x + 1)2 = 7 x + 1 = x = –1 cont’d(x + 1)2 = 7x + 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.
18 Example 3 – Solution Check: x2 + 2x – 6 = 0 cont’dCheck:x2 + 2x – 6 = 0(– )2 + 2 (– ) – 6 ≟ 08 – – – 6 ≟ 08 – 2 – 6 = 0Check the second solution in the original equation.Write original equation.Substitute – for x.Multiply.Solution checks.
19 Completing the SquareWhen 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.
21 The Quadratic FormulaThe 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.”
22 The Quadratic FormulaIn 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.
23 The Quadratic FormulaIf 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.
24 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 = = 45, which is positive. So, the equation has two real solutions.You can solve the equation as follows.x2 + 3x – 9 = 0Write in general form.Quadratic Formula
25 Example 6 – Solution The two solutions are: cont’dThe two solutions are:Check these in the original equation.Substitute a = 1, b = 3,and c = –9.Simplify.Simplify.
26 The Quadratic Formula: One Solution Use the Quadratic Formula to solve 8x2 – 24x + 18 = 0.