The Quadratic Equation A quadratic equation is an equation of the form: One way to solve a quadratic equation is by factoring. Example. Solve 3x2 + 5x – 2 = 0 by factoring. Note that we are using the following property which holds for complex numbers a and b. If ab = 0, then a = 0 or b = 0.

Completing the Square Another method for solving a quadratic equation involves completing the square, which we show by solving 2x2–10x +1 = 0.

The Quadratic Formula By completing the square on the general quadratic, we can obtain the quadratic formula, which is displayed next. The quadratic equation has solutions Example. Solve 2x2–10x +1 = 0. Here, a = 2, b = –10, c = 1. The solution obtained from the quadratic formula is: which agrees with the result we got by completing the square.

The Discriminant The discriminant is the expression b2– 4ac found under the radical in the quadratic formula. If b2– 4ac is negative, we have the square root of a negative number, and the roots are complex conjugate pairs. If b2– 4ac is positive, we have the square root of a positive number, and the roots are two different real numbers. If b2– 4ac = 0, then x = – b/2a. We say that the equation has a double root or repeated root in this case. Example. What does the discriminant tell you about the equation 4x2 –20x + 25 = 0?

Radical Equations If an equation involving x (and no higher powers of x) and a single radical, we proceed as follows to solve the equation. First isolate the radical on one side of the equation. Second, square both sides of the resulting equation to obtain a quadratic equation. Third, solve the quadratic, but be sure to check your answers in the original equation since squaring both sides may have introduced extraneous solutions. Example. Solve

Quadratic Equations and Word Problems We now consider a group of applied problems that lead to quadratic equations. Problem. The larger of two positive numbers exceeds the smaller by 2. If the sum of the squares of the numbers is 74, find the numbers. Solution. Let x = the larger number, x – 2 = the smaller number.

More Quadratic Equations and Word Problems Problem. Working together, computers A and B can complete a data-processing job in 2 hours. Computer A working alone can do the job in 3 hours less than computer B working alone. How long does it take each computer to do the job by itself? Solution. Let x = time for B alone, x – 3 = time for A alone. Then the rate for B is 1/x and the rate for A is 1/(x – 3).

The Quadratic Equation; We discussed The definition of a quadratic equation Solving by factoring Completing the square The quadratic formula The discriminant and its use in predicting the nature of the roots Radical equations and extraneous solutions Word problems for quadratic equations