SECTION A.6 Solving Equations A57 In Problems 95–100, solve each equation by the Square Root Method. 95. = x 25 2 96. = x 36 2 97. ( ) − = x 1 4 2 98. ( ) + = x 2 1 2 99. ( ) + = x2 3 9 2 100. ( ) − = x3 2 4 2 In Problems 101–106, what number should be added to complete the square of each expression? 101. + x x8 2 102. − x x4 2 103. + x x 1 2 2 104. − x x 1 3 2 105. − x x 2 3 2 106. − x x 2 5 2 In Problems 107–112, solve each equation by completing the square. 107. + = x x4 21 2 108. − = x x6 13 2 109. − − = x x 1 2 3 16 0 2 110. + − = x x 2 3 1 3 0 2 111. + − = x x 3 1 2 0 2 112. − − = x x 2 3 1 0 2 In Problems 113–124, find the real solutions, if any, of each equation. Use the quadratic formula. 113. − + = x x4 2 0 2 114. + + = x x4 2 0 2 115. − − = x x5 1 0 2 116. + + = x x5 3 0 2 117. − + = x x 2 5 3 0 2 118. + + = x x 2 5 3 0 2 119. − + = y y 4 2 0 2 120. + + = t t 4 1 0 2 121. = − x x 4 1 2 2 122. = − x x 2 1 2 2 123. + − = x x3 3 0 2 124. + − = x x2 2 0 2 In Problems 125–130, use the discriminant to determine whether each quadratic equation has two unequal real solutions, a repeated real solution, or no real solution without solving the equation. 125. − + = x x5 7 0 2 126. + + = x x5 7 0 2 127. − + = x x 9 30 25 0 2 128. − + = x x 25 20 4 0 2 129. + − = x x 3 5 8 0 2 130. − − = x x 2 3 4 0 2 Problems 131–136 list some formulas that occur in applications. Solve each formula for the indicated variable. 131. Electricity = + R R R R 1 1 1 for 1 2 132. Finance ( ) = + A P rt r 1 for 133. Mechanics = F mv R R for 2 134. Chemistry = PV nRT T for 135. Mathematics = − S a r r 1 for 136. Mechanics = − + v gt v t for 0 139. Find k so that the equation + + = kx x k 0 2 has a repeated real solution. 140. Find k so that the equation − + = x kx 4 0 2 has a repeated real solution. 141. Show that the real solutions of the equation + + = ax bx c 0 2 are the negatives of the real solutions of the equation − + = ax bx c 0. 2 Assume that − ≥ b ac 4 0. 2 142. Show that the real solutions of the equation + + = ax bx c 0 2 are the reciprocals of the real solutions of the equation + + = cx bx a 0. 2 Assume that − ≥ b ac 4 0. 2 137. Show that the sum of the roots of a quadratic equation is − b a . 138. Show that the product of the roots of a quadratic equation is c a . Explaining Concepts 143. Which of the following pairs of equations are equivalent? Explain. (a) = = x x 9; 3 2 (b) = = x x 9; 3 (c) ( )( ) ( ) − − = − − = − x x x x x 1 2 1 ; 2 1 2 144. The equation + + = + + x x x 5 3 3 8 3 has no solution, yet when we go through the process of solving it we obtain = − x 3. Write a brief paragraph to explain what causes this to happen.
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