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Transcript
9-3A Solving Quadratic Equations by Finding Square Roots. Algebra 1 Glencoe McGraw-Hill Linda Stamper You must learn how to use a calculator! There are many makes and models. Read the instruction booklet. Enter a problem into the calculator for which you already know the answer. For example: 4 2nd √ 4 = 2 Keystrokes for TI-30X IIS 4 √ = 2 Keystrokes for TI-30X A Evaluate the expression. Give the exact value, if possible. Otherwise, approximate to the nearest hundredth. You may use a calculator for this section. Example 1 8 2.83 What is the positive square root of 8? Example 2 11 3.32 What is the negative square root of 11? Example 3 27 5.20 What is the positive and negative square root of 27? hundredth s Inverse Operations Recall the use of inverse operations to solve equations. What is the inverse operation of addition? What is the inverse operation of subtraction? What is the inverse operation of multiplication? What is the inverse operation of division? What is the inverse operation of a square number? The inverse of a square number is a square root. A quadratic equation is an equation that can be written in the standard form: ax2 bx c 0 where a 0. a is called the leading coefficient. Quadratic equations can have one solution, two solutions or no real solutions. If b = 0, the equation becomes ax2 c 0 One way to solve a quadratic equation of this form is to isolate the x2 on one side of the equation. Then find the square root(s) of each side. Remember: Squaring a number and finding the square root(s) of a number are inverse operations. Solve the equation. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. x2 4 0 How can you tell this is a 4 4 quadratic equation? 2 x 4 Isolate the square term. x2 4 Undo the square by using square root. x 2 Evaluate the radicals. x 2 and x 2 x 2 Do not give this One of the equations is not solved for a answer!An equation is not positive variable! considered solved if the variable is negative. What do you get when you undo the negative variable? Solve the equation. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Isolate the square term. Undo the square by using square root. Evaluate the radicals. Remember the variable cannot have a + sign because a negative variable is not solved. n2 5 0 5 5 n2 5 n2 5 n 5 n 2.2 Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Example 4 x2 16 0 16 16 x2 16 x2 16 x 4 Example 5 m2 11 0 11 11 m2 11 m2 11 m 11 m 3.3 Remember the variable cannot have a + sign because a negative variable is not solved. Example 6 k 2 100 k2 100 k 10 Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Example 7 3x2 11 11 Example 8 6x2 150 0 Example 9 Example 10 27 3y2 0 x2 15 0 Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Example 7 3x2 11 11 11 11 3x2 0 3 3 x2 0 x2 0 x0 Example 8 6x2 150 0 150 150 6x2 150 6 6 x2 25 x2 25 x 5 Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Example 9 Example 10 x2 15 0 27 3y2 0 15 15 27 27 2 2 3y 27 x 15 3 3 2 x 15 y2 9 15 is undefined y2 9 no real solution y 3 Remember the variable cannot have a + sign because a negative variable is not solved. Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Remember a ± sign indicates two solutions (roots). Factor the P.S.T. x2 18x 81 8 x 92 8 x 92 8 x9 8 x 9 8 There are two solutions (roots): x 9 8 x 9 8 6.2, 11.8 Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Example 11 Example 12 x 4 2 49 y 32 Example 14 Example 15 Example 16 4x2 36 0 x2 10x 25 7 x2 4x 4 5 Hint: Isolate the squared term on one side of the equal sign. 5 Example 13 b2 14b 49 36 Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Example 11 Example 12 y 32 2 x 4 49 x 42 49 x47 x 4 7 x 4 7 or x 4 7 3, 11 y 32 5 5 y 3 5 y 3 5 y 3 5 or y 3 5 5.2, 0.8 Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Example 13 b2 14b 49 36 b 7 2 36 b 7 2 36 b7 6 b76 b 7 6 or b 7 6 13, 1 Example 14 4x2 36 0 4x2 36 4x2 36 2x 6 2x 6 or 2x 6 6 6 x or x 2 2 x 3 Solve the equations. Write the solutions as integers if possible. Otherwise, round to the nearest tenth. Example 15 x2 10x 25 7 2 x 5 7 2 x 5 7 x 5 7 x 5 7 x 5 7 or x 5 7 7.6, 2.4 Example 16 x2 4x 4 5 x 22 5 x 22 5 x 2 5 x 2 5 x 2 5 or x 2 5 0.2, 4.2 9-A5 Handout A5.