Methods of Solving a Quadratic Equation

Q.

Find the roots of the equation $2x^2-5x+3=0$, by factorization.

1. $-\frac{3}{2},1$

2. $4,2$

3. $-\frac{5}{2},\frac{3}{2}$

4. $\frac{3}{2},1$

Q. The sum of all integral values of $k(k\ne 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is

Q. The root(s) of the quadratic equation $2x^2+5x-12=0$ is/are

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

Q. Irrational roots of a quadratic equation with rational coefficients always occurs in conjugate pair.

1. False
2. True

Q.

The number of solutions of the equation $x^2+\left[x\right]-4x+3=0$ is (where $\left[\right]$ denotes G.I.F.)

1. $1$

2. $0$

3. $2$

4. $3$

Q.

Let $f\left(x\right)=x^2+5x+6$, then the number of real roots of $\left(f\right(x){)}^2+5f(x)+6-x=0$ is

1. $2$

2. $1$

3. $3$

4. $0$

Q. For the equation ${\left({\log }_{2}x\right)}^2-4{\log }_{2}x-m^2-2m-13=0,m\in R$.
If the real roots are $x_1,x_2$ such that $x_1<x_2$, then the sum of maximum value of $x_1$ and minimum value of $x_2$ is

1. $\frac{513}{4}$
2. $\frac{257}{4}$
3. $\frac{1025}{8}$
4. $\frac{513}{8}$

Q. Consider the equation $10z^2-3iz-k=0$, where z is a complex variable and $i^2=-1$. Which of the following statements is true?

1. For all purely imaginary numbers $k$, both roots are real and irrational.

2. For all complex numbers $k$, neither root is real.

3. For real positive numbers $k$, both roots are purely imaginary.

4. For real negative numbers $k$, both roots are purely imaginary.

Q. The roots of the quadratic equation $x^2+(p+q)x+pq=0$ are $x=$

1. $-p,-q$
2. $2p,2q$
3. $p,q$

Q.

What is a quadratic equation formula?

Q. The roots of the equation $(a+b-c)x^2-2ax+(a-b+c)=0$ for all $a,b,c\in Q$ are

1. $1,-1$
2. $1,\frac{a-b+c}{a+b-c}$
3. $1,\frac{a+b-c}{a-b+c}$

Q. The quadratic equation $x^2-18x+81=0$ has 2 distinct real root.

1. False
2. True

Q.

The roots of the equation $(3-x{)}^4+(2-x{)}^4=(5-2x{)}^4$ are

1. $2$ real and $2$ non real complex

2. All are non real complex

3. $3$ are real & one non real complex

4. All real

Q. The number of solutions of the equation $\sqrt{4-x}=x-2$ is

Q. The root(s) of the quadratic equation $x^2-8x-12=0$ is/are

1. $4$
2. $6$
3. $2$
4. $8$

Q. The roots of the quadratic equation $p{x}^2+qx+c=0,p\ne 0$ will be given by $x=$

1. $\frac{-q\pm \sqrt{q^2-4pc}}{2p}$
2. $\frac{-p\pm \sqrt{p^2-4qc}}{2p}$
3. $\frac{-p\pm \sqrt{p^2-4qc}}{2q}$

Q. If $p+\sqrt{q}\forall p,q\in Z$ is one of the roots of the quadratic equation, then it's other root will be.

1. $p-\sqrt{q}$
2. $-p-\sqrt{q}$
3. $-p+\sqrt{q}$

Q.

The difference between the two roots of the equation $3{\mathrm{x}}^2+10\mathrm{x}+\mathrm{c}=0$ is $4\frac{2}{3}$ .

Find the solutions for the equation.

Q. The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\cdots \infty }}}}$ is:

1. $2+\frac{4}{\sqrt{5}}\sqrt{30}$
2. $2+\frac{2}{5}\sqrt{30}$
3. $4+\frac{4}{\sqrt{5}}\sqrt{30}$
4. $5+\frac{2}{5}\sqrt{30}$

Q. The linear factor(s) of the equation $x^2{y}^2-9x^2-25y^2+225=0$ is/are

1. $(x+5)=0$
2. $(y+3)=0$
3. $(x-5)=0$
4. $(y-3)=0$

Q. If the quadratic expression $a{x}^2+(a-2+3^{\sqrt{{\log }_{3}5}}-5^{\sqrt{{\log }_{5}3}})x+(5^{{\log }_{5}3}-3^{{\log }_{3}5})$ is negative for exactly two integral values of $x$, then the possible value(s) of $a$ is/are

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

Q. If the sum of a natural number and its reciprocal is $\frac{50}{7}$, then the number is

Q. The zeroes of the polynomial $f\left(x\right)=x^2-3$ is

1. $3,-3$
2. $2,-2$
3. $-\sqrt{3},\sqrt{3}$
4. $0$

Q. For $a,b\in R^{+},$ match the following quadratic equations with their corresponding discriminant.

1. $-23$
2. $4$
3. $-24$
4. $-8$

Q. The sum of all real roots of the equation $\sqrt{5x^2-6x+8}-\sqrt{5x^2-6x-7}=1$ is

1. $\frac{5}{6}$
2. $\frac{6}{5}$
3. $\frac{3}{7}$
4. $\frac{7}{3}$

Q. For $a,b\in R^{+},$ match the following quadratic equations with their corresponding roots.

1. $x=\frac{-1\pm \sqrt{1-4ab}}{2a}$
2. $x=0;\frac{-b}{a}$
3. No Real Roots
4. $x=\frac{-b\pm \sqrt{b^2-4a}}{2}$

Q.

Find the roots of the equation $2x^2+4x+6=0$.

1. $-1\pm \sqrt{5}i$

2. $-1\pm \sqrt{7}i$

3. $-1\pm \sqrt{2}i$

4. $-1\pm \sqrt{3}i$

Q. If $p+\sqrt{q}\forall p,q\in Z$ is one of the roots of the quadratic equation, then it's other root will be.

1. $-p-\sqrt{q}$
2. $-p+\sqrt{q}$
3. $p-\sqrt{q}$

Q. Which of the following is/are the roots of the quadratic equation
$x^2-x-20=0$ ?

1. $x=4$
2. $x=-4$
3. $x=5$
4. $x=-5$

Q. For the quadratic equation $a{x}^2+bx+c=0;a,b,c\in Q$ if it's discriminant $D$ is square of a non zero rational number, then it's roots will be

1. rational & unequal
2. irrational & unequal
3. imaginary & unequal