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The Quadratic Equation

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

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The Quadratic Equation

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  1. 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.

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

  3. The Quadratic Formula • By completing the square on the general quadratic, we can obtain the quadratic formula, which is displayed next. • The quadratic equation hassolutions • 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.

  4. 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?

  5. 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

  6. 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.

  7. 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).

  8. 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

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