SAMPLE College Algebra

110

Chapter 3: Quadratics II and Complex Numbers

• Apply the complete the square method to solve quadratic equations • Discuss imaginary numbers • Graph complex numbers • Solve complex quadratic equations 3.1 The Quadratic Formula A quadratic equation was discussed and defined in Definition 2.10. Recall that a quadratic equation will have the following form. ax 2 + bx + c =0 One can see, just above, that the quadratic equation is set to zero. When a quadratic equation is set equal to zero in this manner, then the quadratic formula can be used to solve it. The use of the quadratic formula is demonstrated hereafter. First, it is worthwhile to practice placing quadratic equations in the form shown, just above. This can involve some of the methods that were discussed in Subsection 1.3 Solving for an Unknown. Solving for an unknown is demonstrated in Solved Problem 1.9. Solved Problem 3.1 Modify the following quadratic equations so that they are set equal to zero. (a) − x 2 +6 x = − 18 (b) 3 x 2 = − 4 x − 2 (c) 5 x 2 = − 6 x − 1 (a) − x 2 +6 x = − 18 A first step is to eliminate -18 from the right side of the equation. Recall that there is a strong association between addition and subtraction. Addition is used to eliminate -18 from the right side. So, 18 is added to both sides of the equation. − x 2 +6 x +18= − 18+18 − x 2 +6 x +18=0 (b) 3 x 2 = − 4 x − 2 A first step is to eliminate -2 from the right side of the equation, by adding 2 to both sides. Then it is necessary to eliminate -4x from the right side of the equation, by adding 4 x to both sides. First, 2 is added to both sides. 3 x 2 = − 4 x − 2 3 x 2 +2= − 4 x − 2+2 3 x 2 +2= − 4 x Then, 4 x is added to both sides of the equation. 3 x 2 +2+4 x = − 4 x +4 x 3 x 2 +4 x +2=0 (c) 5 x 2 = − 6 x − 1 First, 1 is added to both sides. 5 x 2 +1= − 6 x − 1+1 5 x 2 +1= − 6 x