Roots of a Quadratic Equation

Q.

If one zero of the polynomial $x^2-4x+1$ is $(2+\sqrt{3})$ , write the other zero.

Q.

Find the zeros of the following quadratic polynomials.

$t^2-15$

Q.

If one zero of the quadratic polynomial $(k-1)x^2+kx+1$ is -4 then the value of k is

(a) $\frac{-5}{4}$ (b) $\frac{5}{4}$ (c) $\frac{-4}{3}$ (d) $\frac{4}{3}$

Q.

For what value of $k$, $(-4)$ is a zero of the polynomial $x^2-x-(2k+2)$?

Q. If ad $\ne$ bc, then prove that the equation $(a^2+b^2)x^2+2(ac+bd)x+(c^2+d^2)=0$ has no real roots.

Q. Find a quadratic polynomial, the sum and product of whose zeroes are 0 and $-\frac{3}{5}$ respectively.

Q.

Find the zeroes of the polynomial: $p(𝑥)={(𝑥-2)}^2-{(𝑥+2)}^2$.

Q.

If one root of the quadratic equation $2x^2+ax-6=0$ is 2, find the value of a.

1. -1

2. -5

3. 5

4. 3

Q.

A quadratic equation with integral coefficient has integral roots. Justify your answer.

Q. If one root of the equation (k-1)x^2-10x+3=0 is the reciprocal of the other, then the value of k is

Q. If $\alpha$ and $\beta$ are roots of quadratic equation $x^2-1=0$, then find the equation whose roots are $\frac{2\alpha }{\beta }$ and $\frac{2\beta }{\alpha }$

1. $x^2+4x+4$
2. $x^2+4x-4$
3. $x^2-4x+4$
4. $-x^2+4x+4$

Q. If one root of the quadratic equation $3x^2+5a^{2}x+k=0$ is twice the reciprocal of the other root, then the value of $k$ is equal to

1. $3$
2. $6$
3. $5a^2$
4. $1$

Q. If $\alpha \&\beta$ are the zeroes of a quadratic polynomial p(x), and k is any constant, then what is the general form of the polynomial?

1. $k\left[x^2-(\alpha +\beta )x+\alpha \beta \right]$
2. $k{x}^2-(\alpha +\beta )x-\alpha \beta$
3. $k{x}^2+(\alpha +\beta )x-\alpha \beta$
4. $k^2{x}^2+k(\alpha +\beta )x+k\alpha \beta$

Q.

Find the roots of $2x^2-5x+3$.

Q. The zeroes of the polynomial $x^2-x-12$ are -3 and .

1. 4
2. 3
3. -3
4. -4

Q. Let α and β be the roots of equation $x^2-7x-3=0.\text{If}{a}_n={\alpha }_n-{\beta }_{n}forn\ge 1,\text{then value of}\frac{a_{10}-3a_8}{3a_9}$ is

1. $\frac{7}{3}$
2. $\frac{-7}{3}$
3. $\frac{3}{7}$
4. $\frac{-3}{7}$

Q. If the sum of coefficient of ax^2+bx+c=0 is zero (ie. a+b+c=0) then the roots of the equation are: (a) 1 & b/a (b) 1 & c/a (c) 0 & c/a (d) -1

Q.

What are integral roots?

Q. If -2 & -4 are the zeroes of a quadratic polynomial whose leading coefficient is 2, then find the value of (product of zeroes $\times$ coefficient of x)

1. 96
2. -96
3. 16
4. -16

Q.

Find the discriminant of the following quadratic equations $x^2$ – x + 1 = 0

Q.

In each of the following find the value of k for which the given value is a soluton of the given equation
$\left(i\right)7x^2+kx-3=0,x=\frac{2}{3}\left(ii\right)x^2-x(a+b)+k=0,x=a\left(iii\right)k{x}^2+\sqrt{2x}-4=0,x=\sqrt{2}\left(iv\right)x^{2}3ax+k=0,x=-a$

Q. Solve: $\frac{1}{18-x}-\frac{1}{18+x}=\frac{1}{24}$ and $x>0$.

Q.

Show that the roots of the equation $x^2+px-q^2=0$ are real for all real values of p and q

Q.

If x=-3 and $x=\frac{2}{3}$ are solutions of quadratic equation $m{x}^2+7x+n=0$, find the values of m and n.

Q. Question 2 (ii)
Find the roots of the following equations:
(ii) $2x^2+x-4=0$

Q.

In each of the following determine whether the given values are solution of the given equation or not
$\left(i\right)x^2-3x+2=0,x=2,x=-1\left(ii\right)x^2+x+1=0,x=0,x=1\left(iii\right)x^2-3\sqrt{3x}+6=0,x=\sqrt{3},x=-2\sqrt{3}\left(iv\right)x+\frac{1}{x}=\frac{13}{6},x=\frac{5}{6},x=\frac{5}{6},x=\frac{4}{3}\left(v\right)2x^2-x+9=x^2+4x+3,x=2,x=3\left(vi\right)x^2-\sqrt{2x}-4=0,x=-\sqrt{2},x=-2\sqrt{2}\left(vii\right)a^2{x}^2-3abx+2b^2=0,x=\frac{a}{b},x=\frac{b}{a}$

Q.

What term is ${\mathrm{b}}^2-4\mathrm{ac}$?

Q. The perpendicular AD on the base BC ofABCintersects BC in D such that BD=3DC.Prove that $2A{B}^2=2A{C}^2+B{C}^2$

Q.

Determine whether x=-1 is a root of the equation $x^2-3x+2=0$ or not.

Q.

If one root of the equation $2x^2+ax+6=0$ is 3, then the value of a is ___________.

1. -8

2. 7

3. 3

4. 5