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.

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Completing the Square• Another method for solving a quadratic equation involves

completing the square, which we show by solving 2x2–10x +1 = 0.

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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.

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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 .42 xx

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.

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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).

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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