1. From the equation whose roots are
1. 3,-2

2. -5,4

3. $\sqrt{3},-\sqrt{3}$

4. $\frac{1}{2} (-1+\sqrt{5}),\frac{1}{2} (-1-\sqrt{5})$

5. -3+5i,-3-5i

6. a+ib,a-ib

1. Find a quadratic equation whose roots are twice the roots of $4x^2+8x-5=0$

2. Find a quadratic equation whose roots are reciprocals of the roots of $3x^2-5x-2=0$

3. Find a quadratic equation whose roots are greater by h than the roots of $x^2-px+q=0$

4. Find a quadratic equation whose roots are the squares of the roots of $3x^2-5x-2=0$

2. Find a quadratic equation with rational coefficients one of whose roots is
1. 4+3i

2. $\frac{1}{5+3i}$

3. $2+\sqrt{3}$

3. Find the value of k so that the equation
1. $2x^2+kx-15=0$ has one root 3

2. $3x^2+kx-2=0$ has roots whose sum is equal to 6

3. $2x^2+(4-k)x-17=0$ has roots equal but opposite in sign

4. $3x^2+(5+k)x+8=0$ has roots numerically equal but opposite in sign

5. $3x^2+7x+6-k=0$ has one root equal to zero

6. $4x^2-17x+k=0$ has the reciprocal roots

7. $4x^2+kx+5=0$ has roots whose difference is $\frac{1}{4}$

4. Show that -1 is a root of the equation $(a+b-2c)x^2+(2a-b-c)x+(c+a-2b)=0$. Find the other root.

5. Find the value of m for which the equation $(m+1)x^2+2(m+3)x+(2m+3)=0$ will have (a) reciprocal roots (b) one root zero.

6. If the roots of the equation $x^2+ax+c=0$ differ by 1, prove that $a^2=4c+1$

7. If $\alpha, \beta$ are the roots of the equation $x^2-x-6=0$, find the equation whose roots are
1. $\alpha ^2 \beta ^{-1}$and $\beta ^2 \alpha ^{-1}$

2. $\alpha + \frac{1}{\beta}$ and $\beta + \frac{1}{\alpha}$

8. If $\alpha, \beta$ are the roots of the equation $ax^2+bx+c=0$, find the equation whose roots are
1. $\alpha \beta ^{-1}$and $\beta \alpha ^{-1}$

2. $\alpha ^3$ and $\beta ^3$

3. $(\alpha-\beta)^2$ and $(\alpha+\beta)^2$

4. the reciprocal of the roots of given equation

1. If the roots of the equation $ax^2+bx+c=0$ be in the ratio of 3:4, prove that $12b^2=49ac$

2. If one root of the equation $ax^2+bx+c=0$ be four times the other root, show that $4b^2=25ac$

3. For what values of m, the equation $x^2-mx+m+1=0$ may have its root in the ratio 2:3

1. If $\alpha, \beta$ are the roots of the equation $px^2+qx+q=0$, prove that $\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{q}{p}}=0$

2. If roots of the equation $lx^2+nx+n=0$ be in the ratio of p:q, prove that $\sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{n}{l}}=0$

9. If one root of the equation $ax^2+bx+c=0$ be square of the other root, prove that $b^3+a^2c+ac^2=3abc$