Quadratic Equations

Solving Quadratic Equations by the Square Root Property

Martin-Gay, Developmental Mathematics Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce additional methods for solving quadratic equations. Square Root Property If b is a real number and a2 = b, then Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics Square Root Property Example Solve x2 = 49 Solve 2x2 = 4 x2 = 2 Solve (y – 3)2 = 4 y = 3 ± 2 y = 1 or 5 Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics Square Root Property Example Solve x2 + 4 = 0 x2 = −4 There is no real solution because the square root of −4 is not a real number. Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics Square Root Property Example Solve (x + 2)2 = 25 x = −2 ± 5 x = −2 + 5 or x = −2 – 5 x = 3 or x = −7 Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics Square Root Property Example Solve (3x – 17)2 = 28 3x – 17 = Martin-Gay, Developmental Mathematics

Solving Quadratic Equations by the Quadratic Formula

Martin-Gay, Developmental Mathematics The Quadratic Formula Another technique for solving quadratic equations is to use the quadratic formula. The formula is derived from completing the square of a general quadratic equation. Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics The Quadratic Formula A quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions. Martin-Gay, Developmental Mathematics

Solving Quadratic Equations Steps in Solving Quadratic Equations If the equation is in the form (ax+b)2 = c, use the square root property to solve. If not solved in step 1, write the equation in standard form. Try to solve by factoring. If you haven’t solved it yet, use the quadratic formula. Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics The Quadratic Formula Example Solve 11n2 – 9n = 1 by the quadratic formula. 11n2 – 9n – 1 = 0, so a = 11, b = -9, c = -1 Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics The Quadratic Formula Example Solve x2 + x – = 0 by the quadratic formula. x2 + 8x – 20 = 0 (multiply both sides by 8) a = 1, b = 8, c = -20 Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics Solving Equations Example Solve the following quadratic equation. Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics The Quadratic Formula Example Solve x(x + 6) = -30 by the quadratic formula. x2 + 6x + 30 = 0 a = 1, b = 6, c = 30 So there is no real solution. Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics The Discriminant The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant. The discriminant will take on a value that is positive, 0, or negative. The value of the discriminant indicates two distinct real solutions, one real solution, or no real solutions, respectively. Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics The Discriminant Example Use the discriminant to determine the number and type of solutions for the following equation. 5 – 4x + 12x2 = 0 a = 12, b = –4, and c = 5 b2 – 4ac = (–4)2 – 4(12)(5) = 16 – 240 = –224 There are no real solutions. Martin-Gay, Developmental Mathematics

Martin-Gay, Developmental Mathematics Solving Equations Example Solve 3x = x2 + 1. 0 = x2 – 3x + 1 Let a = 1, b = -3, c = 1 Martin-Gay, Developmental Mathematics