50. Quadratic Equations Problems Part 3

101. If \(\alpha, \beta\) are the roots of equation \(x^2 - px + q = 0,\) show that \(\alpha + \frac{1}{\beta}\) is a root of equation \(qx^2 - p(1 + q)x + (1 + q)^2 = 0\)

102. Determine the value of \(m\) for which \(3x^2 + 4mx + 2 = 0\) and \(2x^2 + 3x -2 = 0\) may have a common root.

103. Find the value of \(a\) if \(x^2 - 11x + a = 0\) and \(x^2 - 14x + 2a = 0\) have a common root.

104. If the equations \(ax^2 + bx + c = 0\) and \(bx^2 + cx + a = 0\) have a common root, then show that either \(a + b + c = 0\) or \(a = b = c\)

105. Find the value of \(m\) so that equations \(x^2 + 10x + 21 = 0\) and \(x^2 + 9x + m = 0\) may have a common root. Find also the equation formed by the other roots.

106. Show that the equations \(x^2 - x - 12 = 0\) and \(3x^2 + 10x + 3 = 0\) have a common root. Also find the common root.

107. If the equations \(3x^2 + px + 1 = 0\) and \(2x^2 + qx + 1 = 0\) have a common root, show that \(2p^2 + 3q^2 - 5pq + 1 = 0\)

108. Show that the equation \(ax^2 + bx + c = 0\) and \(x^2 + x + 1 = 0\) cannot have a common root unless \(a = b = c\)

109. If the equations \(x^2 + px + q = 0\) and \(x^2 + p_1x + q_1 = 0\) have a common root, show that it must be either \(\frac{pq_1 - p_1q}{q - q_1}\) or \(\frac{q - q_1}{p_1 - p}\)

110. Prove that the two quadratic equations \(ax^2 + bx + c = 0\) and \(2x^2 - 3x + 4 = 0\) cannot have a common root unless \(6a = -4b = 3c\)

111. Prove that the equations \((q - r)x^2 + (r - p)x + p - q = 0\) and \((r - p)x^2 + (p - q)x + q - r = 0\) have a common root.

112. If the equations \(x^2 + abx + c = 0\) and \(x^2 + acx + b = 0\) have a common root, prove that their other roots satisfy the equation \(x^2 - a(b + c)x + a^2bc = 0\)

113. If the equations \(x^2 - px + q = 0\) and \(x^2 - ax + b = 0\) have a common root and the other root of the second equation is the reciprocal of the other root of the first, then prove that \((q - b)^2 = bq(p - a)^2\)

114. Show that \((x - 2)(x - 3) - 8(x - 1)(x - 3) + 9(x - 1)(x - 2) = 2x^2\) is an identity.

115. Show that \(\frac{a^2(x - b)(x - c)}{(a - b)(a - c)} + \frac{b^2(x - a)(x - c)}{(b - a)(b - c)} + \frac{c^2(x - a)(x - b)}{(c - a)(c - b)} = x^2\) is an identity.

116. Show that \(3x^{10} - 2x^5 + 8 = 0\) is an equation.

117. Solve the equation \(\frac{x + 2}{x - 2} - \frac{x - 2}{x + 2} = \frac{5}{6}\)

118. Solve the equation \(\frac{2\sqrt{x} + 1}{3 - \sqrt{x}} = \frac{11 - 3\sqrt{x}}{5\sqrt{x} - 9}\)

119. Solve the equation \((x + 1)(x + 2)(x - 3)(x - 4) = 336\)

120. Solve the equation \(\sqrt{x + 1} + \sqrt{2x - 5} = 3\)

121. Solve the equation \(2^{2x} + 2^{x + 2} - 32 = 0\)

122. A pilot flies an aircraft with a certain speed a distance of 800 km. He could have saved 40 minutes by increasing the average speed of the aircraft by 40 km/hour. Find the average speed of the aircraft.

123. The length of a rectangle is 2 meter more than its width. If the length is increased by 6 meter and the width is decreased by 2 meter the area becomes 119 square meter. Find the dimensions of the original rectangle.

124. Find the range of values of \(x\) for which \(-x^2 + 3x + 4 > 0\)

125. Find all integral values of \(x\) for which \(5x - 1 < (x + 1)^2 < 7x - 3\)

126. Find all values of \(x\) for which inequality \(\frac{8x^2 + 16x - 51}{(2x - 3)(x + 4)} > 3\) holds.

127. Show that the expression \(\frac{x^2 - 3x + 4}{x^2 + 3x + 4}\) lies between \(7\) and \(\frac{1}{7}\) for real values of \(x\).

128. If \(x\) be real, prove that the expression \(\frac{x^2 + 34x - 71}{x^2 + 2x - 7}\) has no value between \(5\) and \(9\).

129. If \(x\) be real, show that the expression \(\frac{4x^2 + 36x + 9}{12x^2 + 8x + 1}\) can have any real value.

130. Prove that if \(x\) is real, the expression \(\frac{(x - a)(x - c)}{(x - b)}\) is capable of assuming all values if \(a > b > c\) or \(a < b < c\).

131. If \(x + y\) is constant, prove that \(xy\) is maximum when \(x = y.\)

132. If \(x\) be real find the maximum value of \(3 - 6x - 8x^2\) and the corresponding value of \(x.\)

133. Prove that \(\left|\frac{12x}{4x^2 + 9}\right| \le 1\) for all real values of \(x\) or the equality being satisfied only if \(|x| = \frac{3}{2}\)

134. Prove that if the equation \(x^2 + 9y^2 - 4x + 3 = 0\) is satisfied for real values of \(x\) and \(y, x\) must lie between \(1\) and \(3\) and \(y\) must lie between \(-\frac{1}{3}\) and \(\frac{1}{3}\).

135. Find the value of \(a\) for which \(x^2 - ax + 1 - 2a^2 > 0\) for all real values of \(x\).

136. Determine \(a\) such that \(x^2 - 11x + a\) and \(x^2 - 14x + 2a\) may have a common factor.

137. Find the condition that the expression \(ax^2 + bxy + cy^2\) and \(a_1x^2 + b_1xy + c_1y^2\) may have factors \(y - mx\) and \(my - x\) respectively.

138. Find the values of \(m\) for which the expression \(2x^2 + mxy + 3y^2 - 5y - 2\) can be resolved in two linear factors.

139. If the expression \(ax^2 + by^2 + cz^2 + 2ayz + 2bzx + 2cxy\) can be resolved into two rational factors, prove that \(a^3 + b^3 + c^3 = 3abc.\)

140. Find the linear factors of \(2x^2 - y^2 - x + xy + 2y - 1\)

141. Show that the expression \(x^2 + 2(a + b + c)x + 3(bc + ca + ab)\) will be a perfect square if \(a = b = c.\)

142. If \(x\) is real prove that \(2x^2 - 6x + 9\) is always positive.

143. Prove that \(8x - 15 - x^2 > 0\) for limited values of \(x\) and also find the limits.

144. Find the range of the values of \(x\) for which \(-x^2 + 5x - 4 > 0.\)

145. Find the range of the values of \(x\) for which \(x^2 + 6x - 27 > 0.\)

146. Find the solution set of inequation \(\frac{4x}{x^2 + 3}\ge 1, x \in R.\)

147. Find the real values of \(x\) which satisfy \(x^2 - 3x + 2 > 0\) and \(x^2 - 3x - 4 \le 0\)

148. If \(x\) be real and the roots of the equation \(ax^2 + bx + c = 0\) are imaginary, prove that \(a^2x^2 + abx + ac\) is always positive.

149. Prove that the expression \(\frac{x^2 - 2x + 4}{x^2 + 2x + 4}\) lies between \(\frac{1}{3}\) and \(3\) for real values of \(x\).

150. If \(x\) be real show that \(\frac{2x^2 - 3x + 2}{2x^2 + 3x + 2}\) lies between \(7\) and \(\frac{1}{7}\).