# 46. Quadratic Equations Problems Part 1

For what values of \((1 + m)x^2 - 2(1 + 3m)x + (1 + 8m) = 0\) has equal roots?

If \(a + b + c = 0\) and \(a, b, c\) are rational. Prove that the roots of the equation \((b + c - a)x^2 + (c + a - b)x + (a + b - c) = 0\) are rational.

Show that if the roots of the equation \((a^2 + b^2)x^2 + 2x(ac + bd) + c^2 + d^2 = 0\) are real, they will be equal.

If the roots of the equation \(a(b - c)x^2 + b(c - a)x + c(a - b) = 0\) be equal, prove that \(a, b, c\) are in H. P.

If \(a + b + c = 0\) and \(a, b, c\) are real, then prove that equation \((b - x)^2 - 4(a - x)(c - x) = 0\) has real roots and roots will not be equal unless \(a = b = c.\)

Show that if \(p, q, r, s\) are real numbers and \(pr = 2(q + s)\) then at least one of the equations \(x^2 + px + q = 0\) and \(x^2 + rx + s = 0\) has real roots.

If the equation \(x^2 - 2px + q = 0\) has two equal roots, then the equation \((1 + y)x^2 - 2(p + y)x + (q + y) = 0\) will have its roots real and distinct only when \(y\) is negative and \(p\) is not unity.

If the equation \(ax^2 + 2bx + c = 0\) has real roots \(a, b, c\) being real numbers and if \(m\) and \(n\) are real numbers such that \(m^2 > n^2 > 0\) then prove that the equation \(ax^2 + 2mbx + nc = 0\) has real roots.

If the equations \(ax + by = 1\) and \(cx^2 + dy^2 = 1\) have only one solution, prove that \(\frac{a^2}{c} + \frac{b^2}{d} = 1\) and \(x = \frac{a}{c}, y = \frac{b}{d}\)

If \(r\) be the ratio of the roots of the equation \(ax^2 + bx + c = 0,\) show that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\)

If one root of the eq. \((l - m)x^2 + lx + 1 = 0\) be double of the other and if \(l\) be real, show that \(m \leq \frac{9}{7}.\)

If one root of the quadratic equation \(ax^2 + bx + c = 0\) is equal to the nth power of the other, then show that

\[(ac^n)^{\frac{1}{n + 1}} + (a^nc)^{\frac{1}{n + 1}} + b = 0\]If the roots of the equation \(ax^2 + cx + c = 0\) be in the ratio \(p:q,\) show that

\[\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{c}{a}} = 0\]If \(\alpha\) and \(\beta\) be the roots of the equation \(x^2 + px + q = 0\). Find the value of the following in terms of \(p\) and \(q\).

\(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}\)

\((\omega\alpha + \omega^2\beta)(\omega^2\alpha + \omega\beta),\) where \(\omega\) an imaginary cube root of unity.

If \(\alpha\) and \(\beta\) be the roots of the equation \(A(x^2 + m^2) + Amx + cm^2x^2 = 0,\) prove that \(A(\alpha^2 + \beta^2) + A\alpha\beta + c\alpha^2\beta^2 = 0\)

If \(\alpha\) and \(\beta\) be the roots of the equation \(ax^2 + bx + c = 0,\) prove that \(a\left(\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha}\right) + b\left(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right) = b\)

If \(a\) and \(b\) are the roots of the equation \(x^2 + px + 1 = 0\) and \(c, d\) are the roots of the equation \(x^2 + qx + 1 = 0,\) show that \(q^2 - p^2 = (a - c)(b - c)(a + d)(b + d)\)

If the roots of the equation \(x^2 + px + q = 0\) differ from the roots of the equation \(x^2 + qx + p = 0\) by the same quantity, show that \(p + q + 4 = 0.\)

If \(\alpha, \beta\) are the roots of the equation \(ax^2 + bx + c = 0\) and \(S_n = \alpha^2 + \beta^n,\) show that \(aS_{n + 1} + bS_n + cS_{n - 1} = 0\) and hence find \(S_5.\)

If the sum of roots of the equation \(ax^2 + nx + c = 0\) is equal to the sum of the squares of their reciprocals, show that \(bc^2, ca^2, ab^2\) are in A. P.

If \(\alpha\) and \(\beta\) be the values of \(x\) obtained from the equation \(m^2(x^2 - x) + 2mx + 3 = 0\) and if \(m_1\) and \(m_2\) be the two values of \(m\) for which \(\alpha\) and \(\beta\) are connected by the relation \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{4}{3},\) find the value of \(\frac{m_1^2}{m_2} + \frac{m_2^2}{m_1}\)

If the ratio of the roots of the equation \(ax^2 + bx + c = 0\) be equal to the roots of equation \(a_1x^2 + b_1x + c_1 = 0,\) prove that \(\left(\frac{b}{b_1}\right)^2 = \frac{ca}{c_1a_1}\)

Find the quadratic equation with rational coefficients one of whose roots is \(\frac{1}{2 + \sqrt{5}}\).

If \(\alpha\) and \(\beta\) be the roots of the equation \(ax^2 + bx + c =0\) find the quadratic equation whose roots are \(\frac{1}{a\alpha + b}\) and \(\frac{1}{a\beta + b}\).

If \(c, d\) are the roots of the equation \((x - a)(x - b) - k = 0\) show that \(a, b\) are the roots of the equation \((x - c)(x - d) + k = 0\).

The coefficient of \(x\) in the equation \(x^2 + px + q = 0\) was wrongly written as \(17\) in place of \(13\) and roots found to be were \(-2\) and \(-15\). Find the roots of the correct equation.

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

If \(x^2 - ax + b = 0\) and \(x^2 - px + q = 0\) have a common root and the second equation has equal roots, show that \(b + q = \frac{ap}{2}\)

If \(ax^2 + 2bx + c = 0\) and \(a_1x^2 + 2b_1x + c_1 = 0\) have a common root and \(\frac{a}{a_1}, \frac{b}{b_1}, \frac{c}{c_1}\) are in A. P., show that \(a_1, b_1, c_1\) are in G. P.

If each pair of the following three equations \(x^2 + p_1x + q_1 = 0, x^2 + p_2x +q_2 = 0, x^2 + p_3x + q_3 = 0\) has exactly one root in common, then show that \((p_1 + p_2 + p_3)^2 = 4(p_1p_2 + p_2p_3 + p_3p_1 - q_1 - q_2 - q_3)\)

If the equations \(x^2 + cx + cb = 0\) and \(x^2 + bx + ca = 0\) have a common root, show that \(a + b + c = 0;\) show that their other roots are given by the equation \(x^2 + ax + bc = 0\)

If \(a, b, c \in R\) and equations \(ax^2 + bx + c = 0\) and \(x^2 + 2x + 9 = 0\) have a common root, show that \(a : b : c = 1 : 2 : 9\)

Find the value of \(p\) if the equation \(3x^2 - 2x + p = 0\) and \(6x^2 - 17x + 12 = 0\) have a common root.

Show that \(|x|^2 - |x| - 2 = 0\) is an equation.

Show that \(\frac{(x + b)(x + c)}{(b - a)(c - a)} + \frac{(x + c)(x + a)}{(c - b)(a - b)} + \frac{(x + a)(x + b)}{(a - c)(b - c)} = 1\) is an identity.

If \(a, b, c, a_1, b_1, c_1\) are rational and equations \(ax^2 + 2bx + c = 0\) and \(a_1x^2 + 2b_1x + c_1 = 0\) have one and only one root in common, prove that \(b^2 - ac\) and \(b_1^2 - a_1c_1\) must be perfect squares.

If \((a^2 - 1)x^2 + (a - 1)x + a^2 - 4a + 3 = 0\) be an identity in \(x,\) then find the value of \(a\).

Solve: \(\left(x + \frac{1}{x}\right)^2 = 4 + \frac{3}{2}\left(x + \frac{1}{x}\right)\)

Solve the equation: \((x + 4)(x + 7)(x + 8)(x + 11) + 20 = 0\)

Solve the equation \(3^{2x + 1} + 3^2 = 3^{x + 3} + 3^x.\)

Solve for \(x:\) \((5 + 2\sqrt{6})^{x^2 - 3} + (5 - 2\sqrt{6})^{x^2 - 3} = 10\)

A car travels \(25\) km per hour faster than a bus for a journey of \(500\) km. The bus takes 10 hours more than the car. Find the speed of the bus and the car.

Show that the roots of the equation \((a + b)^2x^2 - 2(a^2 - b^2)x + (a - b)^2 = 0\) are equal.

Show that the equation \(3x^2 + 7x + 8 = 0\) cannot be satisfied by any real value of \(x\).

For what value of \(a\) will the roots of the equation \(3x^2 + (7 + a) + 8 - a = 0\) be equal.

If the roots of the equation \((a^2 + b^2)x^2 + 2(ac + bd)x + (c^2 + d^2) = 0\) are equal, then show that \(a : b = c : d\)

Prove that the roots of the equation \((b - c)x^2 + 2(c - a)x + (a - b) = 0\) are always real.

Show that the roots of the equation \(\frac{1}{x - a} + \frac{1}{a} + \frac{1}{x - 1} = 0\) are real for all real values of \(a\).

Show that if \(a + b + c = 0,\) the roots of the equation \(ax^2 + bx + c = 0\) are rational.

Prove that the roots of the equation \((b + c -2a)x^2 + (c + a - 2b)x + (a + b - 2c) = 0\) are rational.