Quadratic Equations with Exactly One Root Common

Q.

Find the roots of quadratic equation $\frac{1}{x}-\frac{1}{x-2}=3$.

Q.

If $\alpha ,\beta \in R$ are such that $1–2i(i^2=–1)$ is a root of $z^2+\alpha z+\beta =0$, then $(\alpha –\beta )$ is equal to:

1. $7$

2. $-3$

3. $3$

4. $-7$

Q.

A value of b for which the equations $x^2+bx-1=0,x^2+x+b=0$ have one root in common, is

1. $-\sqrt{2}$

2. $\pm \sqrt{3}i$

3. $i\sqrt{5}$

4. $\sqrt{2}$

Q.

If $\alpha +\beta =-2$ and ${\alpha }^3+{\beta }^3=-56$, then the quadratic equation, whose roots are $\alpha$ and $\beta$, is

1. $x^2+2x–16=0$

2. $x^2+2x–15=0$

3. $x^2+2x–12=0$

4. $x^2+2x–8=0$

Q.

If the two equations $x^2-cx+d=0$ and $x^2-ax+b=0$ have one common root and the second has equal roots, then $2(b+d)=$)

1. a+c

2. ac

3. 0

4. -ac

Q. The equations $x^2+ax+b=0$ and $x^2+cx+d=0$ have a common root. If the roots of $x^2+ax+b=0$ are equal, then which one of the following options is/are always true?

1. $2(b+d)=ac$
2. $2(a+c)=bd$
3. $c^2=4d$
4. $a^2=4b$

Q. The sum of the values of '$m$' for which the equations $3x^2+4mx+2=0$ and $2x^2+3x-2=0$ may have a common root, is

1. $\frac{3}{8}$
2. $\frac{5}{4}$
3. $\frac{3}{4}$
4. $\frac{5}{8}$

Q. Find the value of $m$ so that, the quadratic equations, $3x^2-2mx-4=0$ and $x(x-4m)+2=0$ have a common root.

1. $-\frac{1}{\sqrt{2}}$
2. $\frac{1}{\sqrt{2}}$
3. $-\sqrt{2}$
4. $\sqrt{2}$

Q. The value of $a+b$ if the equation $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root is

Q.

Two distinct polynomial f(x) and g(x) are defined as follows:

$f\left(x\right)=x^2+ax+2;g\left(x\right)=x^2+2x+a$

If the equation f (x) = 0 and g(x) = 0 have a common root, then the sum of the roots of the equation f (x) + g(x) = 0 is

1. 1

2. $-\frac{1}{2}$

3. 0

4. $\frac{1}{2}$

Q. If $x^2+x+1=0$ and $x^2+ax+b=0$ have a common root, then the minimum value of $(x-a{)}^2+2b$ is

1. $0$
2. $3$
3. $2$
4. $-2$

Q.

If the equation $x^2+px+q=0$ and $x^2+qx+p=0,$ have a common root, where $p\ne q,$ then find the value of $p+q+1?$

1. $0$

2. $-1$

3. $1$

4. $2$

Q. If one root of $a{x}^2+bx+c=0$ is reciprocal of one root of $a_1{x}^2+b_{1}x+c_1=0$, then which of the following condition is correct?

1. $(a{a}_1-c{c}_1{)}^2=(a_{1}b-b_{1}c\left)\right(b_{1}a-b{c}_1)$
2. $(a{a}_1-c{c}_1)=(a_{1}b-b_{1}c)(b_{1}a-b{c}_1)$
3. $(a_{1}b-b_{1}c{)}^2=(a{a}_1-c{c}_1\left)\right(b_{1}a-b{c}_1)$
4. $(b_{1}a-b{c}_1{)}^2=(a{a}_1-c{c}_1\left)\right(a_{1}b-b_{1}c)$

Q. If every pair among the equations $x^2+ax+bc=0,x^2+bx+ca=0$ and $x^2+cx+ab=0$ have a common root, then which of the following is/are correct?

1. Sum of the common roots is $-\frac{1}{2}(a+b+c)$
2. Sum of the common roots is $\frac{1}{2}(a+b+c)$
3. Product of the common roots is $-abc$
4. Product of the common roots is $abc$

Q. The positive value of $\lambda$ for which the equations $x^2-x-12=0$ and $\lambda x^2+10x+3=0$ have one root in common, is

Q.

If $x^3+ax+1=0$ and $x^4+a{x}^2+1=0$ have a common root, then $a$ is equal to:

1. $-2$

2. $2$

3. $-1$

4. $1$

Q. If the equations $x^2+b^2=1-2bx$ and $x^2+a^2=1-2ax$ have one and only one common root, then the value of $|a-b|$ is

1. None of these
2. $1$
3. $0$
4. $2$

Q. If $x^2+x+1=0$ and $x^2+ax+b=0$ have a common root, then the minimum value of $(x-a{)}^2+2b$ is

1. $-2$
2. $3$
3. $2$
4. $0$

Q. If $(x-2)$ is common factor of expressions $x^2+ax+b$ and $x^2+cx+d$, then $\frac{b-d}{c-a}=$
$(a\ne c)$

Q.

Let $p\left(x\right)=x^2–5x+a$ and $q\left(x\right)=x^2–3x+b$, where a and b are positive integers. Suppose hof(p(x), q(x)) = x – 1 and k(x) = 1cm (p(x), q(x)). If the coefficient of the highest degree term of k(x) is 1, the sum of the roots of (x – 1) + k(x) is.

1. 7

2. 6

3. 5

4. 4

Q. If $(x-2)$ is common factor of expressions $x^2+ax+b$ and $x^2+cx+d$, then $\frac{b-d}{c-a}=$
$(a\ne c)$

Q. If two quadratic equations
$x^2+bx-1=0\&x^2+x+b=0$ have a root in common, then $b^3+3b=$

1. $1$​​​​​
2. $-1$
3. $0$

Q. If $x^2-11x+a$ and $x^2-14x+2a$ have common factor, then value(s) of $a$ is/are

1. $a=4$
2. $a=0$
3. $a=24$
4. $a=16$

Q.

If the equation $x^2+2x+3\lambda =0$ and $2x^2+3x+5\lambda =0$ have a non - zero common root, then $\lambda$ is equal to:

1. None

2. $3$

3. $-1$

4. $1$

Q. If two quadratic equations
$x^2+bx-1=0\&x^2+x+b=0$ have a root in common, then $b^3+3b=$

1. $1$​​​​​
2. $0$
3. $-1$

Q.

If $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0$ have a common root and $a,b,c$ are non-zero real numbers, then$\frac{a^3+b^3+c^3}{abc}$ is equal to:

1. $2$

2. $3$

3. $1$

4. None

Q. If $\alpha ,\beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^2+2\beta x+\gamma =0$ and $x^2+x–1=0$ have a common root, then $\alpha (\beta +\gamma )$ is equal to :

1. $\alpha \gamma$
2. $\alpha \beta$
3. $\beta \gamma$
4. $0$

Q. If $x^2+ax+10=0$ and $x^2+bx-10=0$ have a common root, then $a^2-b^2$ is equal to

Q. If the two polynomials $x^2-11x+a$ and $x^2-14x+2a$ have a common factor, then $a=$

1. $0$
2. $24$
3. $0,3$
4. $0,24$

Q. The quadratic equations $x^2-6x+a=0,$ $x^2-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4:3$ . Then common root is :

1. $2$
2. $1$
3. $4$
4. $3$