Quadratic Equations, Functions, Zeros, and Models Section 3.2 Quadratic Equations, Functions, Zeros, and Models

Objectives Find zeros of quadratic functions and solve quadratic equations by using the principle of zero products, by using the principle of square roots, by completing the square, and by using the quadratic formula. Solve equations that are reducible to quadratic. Solve applied problems using quadratic equations.

Quadratic Equations A quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, a  0, where a, b, and c are real numbers. A quadratic equation written in this form is said to be in standard form.

Quadratic Functions A quadratic function f is a function that can be written in the form f (x) = ax2 + bx + c, a  0, where a, b, and c are real numbers. The zeros of a quadratic function f (x) = ax2 + bx + c are the solutions of the associated quadratic equation ax2 + bx + c = 0. (These solutions are sometimes called roots of the equation.) Quadratic functions can have real-number zeros or imaginary-number zeros and quadratic equations can have real-number or imaginary-number solutions.

Equation-Solving Principles The Principle of Zero Products: If ab = 0 is true, then a = 0 or b = 0, and if a = 0 or b = 0, then ab = 0.

Equation-Solving Principles The Principle of Square Roots: If x2 = k, then

Example Solve 2x2  x = 3.

Example continued Check: Check: x = – 1 The solutions are –1 and TRUE

Example Solve 2x2  10 = 0. Check: TRUE The solutions are and

Completing the Square To solve a quadratic equation by completing the square: Isolate the terms with variables on one side of the equation and arrange them in descending order. Divide by the coefficient of the squared term if that coefficient is not 1. Complete the square by finding half the coefficient of the first-degree term and adding its square on both sides of the equation. Express one side of the equation as the square of a binomial. Use the principle of square roots. Solve for the variable.

Example Solve 2x2  1 = 3x. The solutions are

Quadratic Formula The solutions of ax2 + bx + c = 0, a  0, are given by This formula can be used to solve any quadratic equation.

Example Solve 3x2 + 2x = 7. Find exact solutions and approximate solutions rounded to the thousandths. 3x2 + 2x  7 = 0 a = 3, b = 2, c = 7 The exact solutions are: The approximate solutions are –1.897 and 1.230.

Discriminant The expression b2  4ac shows the nature of the solutions. This expression is called the discriminant. For ax2 + bx + c = 0, where a, b, and c are real numbers: b2  4ac = 0 One real-number solution; b2  4ac > 0 Two different real-number solutions; b2  4ac < 0 Two different imaginary-number solutions, complex conjugates.

Equations Reducible to Quadratic Some equations can be treated as quadratic, provided that we make a suitable substitution. Example: x4  5x2 + 4 = 0 Knowing that x4 = (x2)2, we can substitute u for x2 and the resulting equation is then u2  5u + 4 = 0. This equation can then be solved for u by factoring or using the quadratic formula. Then the substitution can be reversed by replacing u with x2, and solving for x. Equations like this are said to be reducible to quadratic, or quadratic in form.

Example Solve: x4  5x2 + 4 = 0. x4  5x2 + 4 = 0 u2  5u + 4 = 0 (substituting u for x2) (u  1)(u  4) = 0 u  1 = 0 or u  4 = 0 u = 1 or u = 4 x2 = 1 or x2 = 4 x = ±1 or x = ±2 The solutions are 1, 1, 2, and 2.

Applications: Example Some applied problems can be translated to quadratic equations. Example Magazine Closures. The numbers of both magazine launches and magazine closures have increased in recent years. The function can be used to estimate the number of magazine closures after 2012. In what year was the number of magazine closures 99?

Example continued Solution We substitute 99 for m(x) and solve for x: We then use the quadratic formula, with a = 34, b = −59, and c = −18:

Example continued Because we are looking for a year after 2012, we use the positive solution. Thus there were 99 magazine closures in 2 years after 2012, or in 2014.