Chapter 8

Form an equation whose roots are negative of the roots of the equation \(x^{3}-5 x^{2}-7 x-3=0\).

Form an equation whose roots are double the roots of the equation \(x^{3}+x+1=0\).

Form an equation whose roots exceed the roots of the equation \(x^{3}+x+1=0\) by 2 .

If \(\alpha, \beta\) are the roots of \(a x^{2}+b x+c=0\) then find the equation whose roots are \(2+\alpha, 2+\beta\).

Form an equation whose roots are reciprocal of the roots of the equation \(x^{3}+x+1=0\).

Find the quadratic equation whose roots are reciprocal of the roots of the equation \(a x^{2}+b x+c=0\).

Form an equation whose roots are squares of the roots of the equation \(x^{3}-6 x^{2}+11 x-6=0\).

Form an equation whose roots are cubes of the roots of the equation \(a x^{3}+b x^{2}+c x+d=0\).

If \(\alpha, \beta, \gamma\) are roots of \(x^{3}+x+1=0\), then find the polynomial whose roots are \(\frac{3}{\alpha^{2}+1}, \frac{3}{\beta^{2}+1}, \frac{3}{\gamma^{2}+1}\).

For what values of \(a\), the roots of the equation \(x^{2}+a^{2}=8 x+6 a\) are real.

For what values of \(a\) the function \(x^{2}-a x+1\) has no real roots?

If one root of the equation \(x^{2}+p x+12=0\) is 4, while the equation \(x^{2}+p x+q=0\) has equal roots, then find the value of \(q\).

For what values of \(k, x^{2}-k x+k+2\) has equal roots?

Find the values of \(k\) for which the quadratic \((k+11) x^{2}-(k+3) x+1\) has real and equal roots.

If \(a\) and \(b\) are integers and \(x^{2}+a x+b=0\) has discriminant as a perfect square then prove that its roots are integers.

Show that the expression \(a x^{2}+b x+c\) has always the same sign as \(c\) if \(4 a c>b^{2}\).

Find the values of \(a\) for which \(\left(a^{2}-1\right) x^{2}+2(a-1) x+2\) is positive for any \(x\).

If the graph of the function \(y=16 x^{2}+8(a+5) x-7 a-5\) is strictly above the \(x\) -axis, then find \(a\).

For what values of \(m\) the function \(m x^{2}-9 m x+5 m+1\) is positive for all \(x ?\)

For what values of \(m\) the function \(m x^{2}-9 m x+5 m+1\) is negative for all \(x\) ?

Prove that the roots of \((x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0\) are always real and they will be equal if and only if \(a=b=c\).

Examine the nature of the roots of the quadratic \((b-x)^{2}-4(a-x)(c-x)=0\) where \(a, b, c\) are real.

Show that the equation \(a x^{2}+b x+c=0\) where \(a, b, c\) are real numbers connected by the relation \(4 a+2 b+c=0\) and \(a b>0\) has real roots.

If the roots of the equation \(x^{2}-a x+b=0\) are real and differ by a quantity which is less than \(c(c>0)\), then \(b\) lies between \(\frac{a^{2}-c^{2}}{4}\) and \(\frac{a^{2}}{4}\)

If the two roots of the equation \((a-1)\left(x^{2}+x+1\right)^{2}-(a+1)\left(x^{4}+x^{2}+1\right)=0\) are real and distinct then prove that \(a\) lies in the interval \((-\infty,-2) \cup(2, \infty)\)

Show that a polynomial of an odd degree has at least one real root.

Show that a polynomial of an even degree has at least two real roots if it attains at least one value opposite in sign to the coefficient of its highest degree term.

For what value of \(a, x^{2}-11 x+a=0\) and \(x^{2}-14 x+2 a=0\) have a common root?

If \(x^{2}-a x-21=0\) and \(x^{2}-3 a x+35=0\) have a common root, then find the value of \(a\).

Find \(k\) if the equations \(4 x^{2}-11 x+2 k=0\) and \(x^{2}-3 x-k=0\) have a common root and obtain the common root for this value of \(k\). \\{Ans. \(k=0\) or \(-\frac{17}{36}\), common root 0 or \(\left.\frac{17}{6}\right\\}\)

If the equations \(x^{2}+2 x+3 \lambda=0\) and \(2 x^{2}+3 x+5 \lambda=0\) have a non-zero common root, then find the value of \(\lambda\).

If \(\alpha, \beta\) are the roots of \(x^{2}+p x+q=0\) and \(\gamma, \delta\) are the roots of \(x^{2}+r x+s=0\), evaluate \((\alpha-\gamma)(\alpha-\delta)(\beta-\gamma)(\beta-\delta)\) in terms of \(p, q, r\) and \(s\). Hence deduce the condition that the equations have a common root.

If the equations \(x^{2}+b x+c a=0\) and \(x^{2}+c x+a b=0\) have a common root, then show that their other roots are the roots of the equation \(x^{2}+a x+b c=0\).

Find the condition that a root of the equation \(a x^{2}+b x+c=0\) be reciprocal of a root of the equation \(a^{\prime} x^{2}+b^{\prime} x+c^{\prime}=0 .\)

If each pair of the three equations \(x^{2}+p_{1} x+q_{1}=0, x^{2}+p_{2} x+q_{2}=0\) and \(x^{2}+p_{3} x+q_{3}=0\) have a common root, then prove that \(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+4\left(q_{1}+q_{2}+q_{3}\right)=2\left(p_{1} p_{2}+p_{2} p_{3}+p_{3} p_{1}\right)\)

If every pair of equations \(x^{2}+a x+b c=0, x^{2}+b x+c a=0, x^{2}+c x+a b=0\) have a common root, then find the sum and product of these common roots.

If the three equations \(x^{2}+a x+12=0, x^{2}+b x+15=0\) and \(x^{2}+(a+b) x+36=0\) have a common positive root, then find \(a\) and \(b\) and the roots.

If the roots of the equation \((m-3) x^{2}-2 m x+5 m=0\) are real and positive then prove that \(m \in\left[3, \frac{15}{4}\right]\).

For what values of \(a\) do both roots of the function \(x^{2}-6 a x+\left(2-2 a+9 a^{2}\right)\) exceed 3 ?

For what values of \(a\), the roots of the equation \(x^{2}-2 a x+a^{2}+a-3=0\) are real and less than 3 ?

Find all the values of \(m\) for which both roots of the function \(2 x^{2}+m x+m^{2}-5\) i. are less than 1. ii. exceed \(-1\).

For what values of \(a\) does the function \(x^{2}+2(a+1) x+9 a-5\) has i. no real roots. ii. only negative roots. iii. only positive roots.

If \(b, c>0\), then show that roots of the equation \(x^{2}+b x-c=0\) are of opposite sign.

Find the set of values of \(p\) for which the roots of the equation \(3 x^{2}+2 x+p(p-1)=0\) are of opposite signs.

If the roots of the equation \(3 x^{2}+2\left(k^{2}+1\right) x+\left(k^{2}-3 k+2\right)=0\) be of opposite signs, then prove that \(1

For what values of \(a\) does the function \(2 x^{2}-\left(a^{3}+8 a-1\right) x+a^{2}-4 a\) has the roots of opposite signs?

For what values of \(a\) does the function \(\left(a^{2}-a-2\right) x^{2}+2 a x+a^{3}-27\) has the roots of opposite signs?

For what values of \(a\) do both roots of the function \(x^{2}-a x+2\) belong to the interval \([0,3]\) ?

For what values of \(k\) do both roots of the function \(x^{2}+2(k-3) x+9\) belong to the interval \((-6,1)\) ?

Find all the values of \(k\) for which one root of the function \(x^{2}-(k+1) x+k^{2}+k-8\) exceeds 2 and the other root is less than \(2 ?\)