.. meta::
   :author: Shiv Shankar Dayal
   :title: Quadratic Equations Problems Part 3
   :description: Quadratic Equations Problems Part 3
   :keywords: quadratic equations, algebra

Quadratic Equations Problems Part 3
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101. If :math:`\alpha, \beta` are the roots of equation :math:`x^2 - px + q = 0,` show that :math:`\alpha +
     \frac{1}{\beta}` is a root of  equation :math:`qx^2 - p(1 + q)x + (1 + q)^2 = 0`
102. Determine the value of :math:`m` for which :math:`3x^2 + 4mx + 2 = 0` and :math:`2x^2 + 3x -2 = 0` may have a
     common root.
103. Find the value of :math:`a` if :math:`x^2 - 11x + a = 0` and :math:`x^2 - 14x + 2a = 0` have a common root.
104. If the equations :math:`ax^2 + bx + c = 0` and :math:`bx^2 + cx + a = 0` have a common root, then show that either
     :math:`a + b + c = 0` or :math:`a = b = c`
105. Find the value of :math:`m` so that equations :math:`x^2 + 10x + 21 = 0` and :math:`x^2 + 9x + m = 0` may have a
     common root. Find also the equation formed by the other roots.
106. Show that the equations :math:`x^2 - x - 12 = 0` and :math:`3x^2 + 10x + 3 = 0` have a common root. Also find the
     common root.
107. If the equations :math:`3x^2 + px + 1 = 0` and :math:`2x^2 + qx + 1 = 0` have a common root, show that
     :math:`2p^2 + 3q^2 - 5pq + 1 = 0`
108. Show that the equation :math:`ax^2 + bx + c = 0` and :math:`x^2 + x + 1 = 0` cannot have a common root unless
     :math:`a = b = c`
109. If the equations :math:`x^2 + px + q = 0` and :math:`x^2 + p_1x + q_1 = 0` have a common root, show that it must be
     either :math:`\frac{pq_1 - p_1q}{q - q_1}` or :math:`\frac{q - q_1}{p_1 - p}`
110. Prove that the two quadratic equations :math:`ax^2 + bx + c = 0` and :math:`2x^2 - 3x + 4 = 0` cannot have a common
     root unless :math:`6a = -4b = 3c`
111. Prove that the equations :math:`(q - r)x^2 + (r - p)x + p - q = 0` and :math:`(r - p)x^2 + (p - q)x + q - r = 0`
     have a common root.
112. If the equations :math:`x^2 + abx + c = 0` and :math:`x^2 + acx + b = 0` have a common root, prove that their other
     roots satisfy the equation :math:`x^2 - a(b + c)x + a^2bc = 0`
113. If the equations :math:`x^2 - px + q = 0` and :math:`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 :math:`(q - b)^2 = bq(p - a)^2`
114. Show that :math:`(x - 2)(x - 3) - 8(x - 1)(x - 3) + 9(x - 1)(x - 2) = 2x^2` is an identity.
115. Show that :math:`\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 :math:`3x^{10} - 2x^5 + 8 = 0` is an equation.
117. Solve the equation :math:`\frac{x + 2}{x - 2} - \frac{x - 2}{x + 2} = \frac{5}{6}`
118. Solve the equation :math:`\frac{2\sqrt{x} + 1}{3 - \sqrt{x}} = \frac{11 - 3\sqrt{x}}{5\sqrt{x} - 9}`
119. Solve the equation :math:`(x + 1)(x + 2)(x - 3)(x - 4) = 336`
120. Solve the equation :math:`\sqrt{x + 1} + \sqrt{2x - 5} = 3`
121. Solve the equation :math:`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 :math:`x` for which  :math:`-x^2 + 3x + 4 > 0`
125. Find all integral values of :math:`x` for which :math:`5x - 1 < (x + 1)^2 < 7x - 3`
126. Find all values of :math:`x` for which inequality :math:`\frac{8x^2 + 16x - 51}{(2x - 3)(x + 4)} > 3` holds.
127. Show that the expression :math:`\frac{x^2 - 3x + 4}{x^2 + 3x + 4}` lies between :math:`7` and :math:`\frac{1}{7}`
     for real values of :math:`x`.
128. If :math:`x` be real, prove that the expression :math:`\frac{x^2 + 34x - 71}{x^2 + 2x - 7}` has no value between
     :math:`5` and :math:`9`.
129. If :math:`x` be real, show that the expression :math:`\frac{4x^2 + 36x + 9}{12x^2 + 8x + 1}` can have any real
     value.
130. Prove that if :math:`x` is real, the expression :math:`\frac{(x - a)(x - c)}{(x - b)}` is capable of assuming all
     values if :math:`a > b > c` or :math:`a < b < c`.
131. If :math:`x + y` is constant, prove that :math:`xy` is maximum when :math:`x = y.`
132. If :math:`x` be real find the maximum value of :math:`3 - 6x - 8x^2` and the corresponding value of :math:`x.`
133. Prove that :math:`\left|\frac{12x}{4x^2 + 9}\right| \le 1` for all real values of :math:`x` or the equality being
     satisfied only if :math:`|x| = \frac{3}{2}`
134. Prove that if the  equation :math:`x^2 + 9y^2 - 4x + 3 = 0` is satisfied for real values of :math:`x` and :math:`y,
     x` must lie between :math:`1` and :math:`3` and :math:`y` must lie between :math:`-\frac{1}{3}` and
     :math:`\frac{1}{3}`.
135. Find the value of :math:`a` for which :math:`x^2 - ax + 1 - 2a^2 > 0` for all real values of :math:`x`.
136. Determine :math:`a` such that :math:`x^2 - 11x + a` and :math:`x^2 - 14x + 2a` may have a common factor.
137. Find the condition that the expression :math:`ax^2 + bxy + cy^2` and :math:`a_1x^2 + b_1xy + c_1y^2` may have
     factors :math:`y - mx` and :math:`my - x` respectively.
138. Find the values of :math:`m` for which the expression :math:`2x^2 + mxy + 3y^2 - 5y - 2` can be resolved in two
     linear factors.
139. If the expression :math:`ax^2 + by^2 + cz^2 + 2ayz + 2bzx + 2cxy` can be resolved into two rational factors, prove
     that :math:`a^3 + b^3 + c^3 = 3abc.`
140. Find the linear factors of :math:`2x^2 - y^2 - x + xy + 2y - 1`
141. Show that the expression :math:`x^2 + 2(a + b + c)x + 3(bc + ca + ab)` will be a perfect square if :math:`a = b =
     c.`
142. If :math:`x` is real prove that :math:`2x^2 - 6x + 9` is always positive.
143. Prove that :math:`8x - 15 - x^2 > 0` for limited values of :math:`x` and also find the limits.
144. Find the  range of the values of :math:`x` for which :math:`-x^2 + 5x - 4 > 0.`
145. Find the range of the values of :math:`x` for which :math:`x^2 + 6x - 27 > 0.`
146. Find the solution set of inequation :math:`\frac{4x}{x^2 + 3}\ge 1, x \in R.`
147. Find the real values of :math:`x` which satisfy :math:`x^2 - 3x + 2 > 0` and :math:`x^2 - 3x - 4 \le 0`
148. If :math:`x` be real and the roots of the equation :math:`ax^2 + bx + c = 0` are imaginary, prove that
     :math:`a^2x^2 + abx + ac` is always positive.
149. Prove that the expression :math:`\frac{x^2 - 2x + 4}{x^2 + 2x + 4}` lies between :math:`\frac{1}{3}` and :math:`3`
     for real values of :math:`x`.
150. If :math:`x` be real show that :math:`\frac{2x^2 - 3x + 2}{2x^2 + 3x + 2}` lies between :math:`7` and
     :math:`\frac{1}{7}`.