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Section P7 Equations Solving Linear Equations in One Variable Example Solve the equation: 2(3x-5)=-5(x-1)+x Linear Equations with Fractions Solving with Fractions Example Solve for x. x 1 3 2x 1 3 2 12 Example Solve for x. x 3 x 1 x 2 5 2 5 Rational Equations A rational equation is an equation containing one or more rational expressions. Notice how the variables appear in the denominator in rational equations and the previous examples ( Linear Equations with Fractions) only had variables in the numerator. Solving Rational Equations Example Solve the Rational Equation. 3 4 1 2 x2 x 4 x2 Example 5 2 1 Solve the Rational Equation. x3 x 2 Example 3 2 Solve the Rational Equation. 2 6 x 1 x 1 Solving a Formula for One of Its Variables A-Prt=P A=Prt+P A=P(rt+1) A P or rt+1 A P= rt+1 Example Solve for l in the formula for the Perimeter of a rectangle. Example 1 Solve for b in the formula for the area of a triangle A= bh. 2 Equations Involving Absolute Value Example Solve: x 5 9 7 Example Solve: 2 2x 7 14 0 Quadratic Equations and Factoring Example Solve the equation (2x-5)(3x+4)=0 using the Zero-Product Principle. Example Solve the equation by factoring: x 2 3x 4 0 Example Solve the equation by factoring: 2x 2 7 x 4 0 Quadratic Equations and the Square Root Property Example Solve the following problem by the square root property. (x-4)2 25 Example Solve the following problem by the square root property. 4x 2 7 0 Quadratic Equations and Completing the Square Obtaining a Perfect Square Trinomial Start Add 1 b 2 2 Result 2 x 6x 1 g 6 9 2 x 4x 1 g 4 4 2 x 2 20 x 1 g 20 100 2 2 x2 6x 9 x 3 x2 4x 4 x 2 2 2 Factored Form 2 x 2 20 x 100 2 2 x 10 2 Completing the Square Example Complete the square to solve the following problem. x 2 10 x 3 0 Example Complete the square to solve the following problem. x 2 8 x 13 0 Example Complete the square to solve the following problem. x 2 5 x 10 Quadratic Equations and the Quadratic Formula Example Solve the equation using the quadratic formula. x 2 6 x 3 Example Solve the equation using the quadratic formula. 2x 2 4 x 5 Quadratic Equations and the Discriminant Example Use the discriminant to find the number and types of solutions, but don't solve the equation. a. x 2 5 x 6 0 b. x 2 3 x 9 c. 2x 2 4 x 9 Graphing Calculator The real solutions of a quadratic equation ax2+bx+c=0 correspond to the x-intercepts of the graph. The U shaped graph shown below has two x intercepts. When y=0, the value(s) of x will be the solution to the equation. Since y=0 these are called the zeros of the function. Solving Polynomial Equations using the Graphing Calculator By pressing 2nd Trace to get Calc, then the #2,you get the zeros. It will ask you for left and right bounds, and then a guess. For left and right bounds move the blinking cursor (using the arrow keys-cursor keys) to the left and press enter. Then move the cursor to the right of the x intercept and press enter. Press enter when asked to guess. Then you get the zeros or solution. Repeat this process for each x intercept. Determining Which Method to Use Example Factor and solve. -3x 2 6 x 0 Example Solve by any method. -3x 2 15 0 Example Solve by any method. x 2 4 x 10 0 Radical Equations A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. We solve the equation by squaring both sides. x4 If we square both sides, we obtain x 2 16 x 16 -4 or 4 This new equation has two solutions, -4 and 4. By contrast, only 4 is a solution of the original equation, x=4. For this reason, when raising both sides of an equation to an even power, check proposed solutions in the original equation. Extra solutions may be introduced when you raise both sides of a radical equation to an even power. Such solutions, which are not solutions of the given equation are called extraneous solutions or extraneous roots. Example Solve and check your answers: x 5 x 1 1 Solve for h in the area formula for a trapezoid. A= h (a b) 2 (a) (b) (c) (d) 2A ab A 2( a b) A ab A 2a b Solve: 3 x 8 27 0 (a) x 3, 3 (b) x 4,10 (c) x 1, 17 (d) x 17, 1