Section P.2 Solving Equations Important Vocabulary

Section P.2 Solving Equations 5

Course Number Section P.2 Solving Equations Instructor Objective: In this lesson you learned how to solve linear and nonlinear equations. Date

Important Vocabulary Define each term or concept.

Equation A statement, usually involving x, that two algebraic expressions are equal.

Extraneous solution A solution that does not satisfy the original equation.

Quadratic equation An equation in x that can be written in the general form 2 ax + bx + c = 0 where a, b, and c are real numbers with a ¹ 0.

I. Equations and Solutions of Equations (Page 12) What you should learn How to identify different To solve an equation in x means to . . . find all the values of x types of equations for which the solution is true. The values of x for which the equation is true are called its solutions . An identity is . . . an equation that is true for every real number in the domain of the variable. A conditional equation is . . . an equation that is true for just some (or even none) of the real numbers in the domain of the variable.

II. Linear Equations in One Variable (Pages 12-14) What you should learn A linear equation in one variable x is an equation that can be How to solve linear equations in one variable written in the standard form ax + b = 0 , where a and b and equations that lead to are real numbers with a ¹ 0 . linear equations A linear equation has exactly one solution(s).

To solve a conditional equation in x, . . . isolate x on one side of the equation by a sequence of equivalent, and usually simpler, equations, each having the same solution(s) as the original equation.

An equation can be transformed into an equivalent equation by one or more of the following steps: (1) Remove symbols of grouping, combine like terms, or simplify fractions on one or both sides of the equation.

(2) Add (or subtract) the same quantity to (from) each side of the equation. (3) Multiply (or divide) each side of the equation by the same nonzero quantity. (4) Interchange the two sides of the equation.

Example 1: Solve 5(x + 3) = 35. The solution is 4.

To solve an equation involving fractional expressions, . . . find the least common denominator of all terms and multiply every term by this LCD.

When is it possible to introduce an extraneous solution? When multiplying or dividing an equation by a variable quantity.

5x 9 1 5x 4 Example 2: Solve: (a) = (b) + = 7 14 x +1 x 2 -1 x -1 (a) 0.9 (b) 2.5

III. Quadratic Equations (Pages 15-17) What you should learn How to solve quadratic To solve a quadratic equation by factoring, . . . write the equations by factoring, equation in general form with all terms collected on the left side extracting square roots, and using the Quadratic and zero on the right. Then factor the left side of the quadratic Formula equation as the product of two linear factors. Finally, find the solutions of the quadratic equation by setting each factor equal to zero.

Example 3: Solve x2 -12x = -27 by factoring. The solutions are 3 and 9.

The equation u 2 = c , where c > 0, has exactly two solutions: u = Ö c and u = - Ö c . These solutions can also be written as u = ± Ö c .

Example 4: Solve 5(x - 4) 2 = 45 by extracting square roots. The solutions are 1 and 7.

If ax 2 + bx + c = 0 , then the Quadratic Formula gives the solutions as x = (- b ± Ö b2 - 4ac ) / (2a) .

Example 5: For the quadratic equation 16 - 3x = -2x 2 , find the values of a, b, and c to be substituted into the Quadratic Formula. a = 2, b = - 3, and c = 16, OR a = - 2, b = 3, and c = - 16

IV. Polynomial Equations of Higher Degree (Page 18) What you should learn How to solve polynomial To solve polynomial equations of higher degree, . . . use the equations of degree three same methods as those used to solve quadratic equations, such as or greater factoring, factoring by grouping, extracting square roots, or using the Quadratic Formula.

Example 6: Describe a strategy for solving the polynomial equation x3 + 2x 2 - x = 2 . Then find the solutions. First write the polynomial equation in general form with zero on the right-hand side of the equation. Then factor the polynomial by grouping to solve. The solutions are - 2, - 1, and 1.

Example 7: Solve the equation: x4 - 4x 2 - 45 = 0 The real solutions are - 3 and 3.

V. Equations Involving Radicals (Page 19) What you should learn Operations that can introduce extraneous solutions include, . . . How to solve equations involving radicals squaring each side of an equation, raising each side of an equation to a rational power, and multiplying each side of an equation by a variable quantity.

If any of these operations is performed while solving an equation, . . . then it is necessary to check the resulting solutions in the original equation.

Example 8: Describe a strategy for solving the following equation involving a radical expression: 8 - x -15 = 0 Add 15 to both sides to isolate the radical expression. Then square both sides to eliminate the radical. Finally, solve for x and check the solution in the original equation.

VI. Equations Involving Absolute Values (Page 20) What you should learn How to solve equations To solve an equation involving an absolute value, . . . involving absolute values remember that the expression inside the absolute value signs can be positive or negative, resulting in two separate equations to be solved.

Example 9: Write the two equations that must be solved to solve the absolute value equation 3x 2 + 2x - 5 = 0 . 3x2 + 2x = 5 and - (3x2 + 2x) = 5

Example 10: Solve x 2 + 4x - 5 = 0 . The solutions are -5 and 1.

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