Roots of Quadratic Equation by Factorization

Q.

What is the range of quadratic equations?

Q.

Solve $x^2$ + 8𝒙 - 48 = 0 by completing the square method. ( 2 marks)

Q.

Is a parabola a function?

Q.

The length of the sides of a right triangle is $5x+2,5x$ and $3x-1$. Find the length of each side.

Q.

Divide $16$into two parts such that twice the square of the larger part exceeds the square of the smaller part by $164.$

Q. Find the common root of the given quadratic equations.
$x^2-4=0and{x}^2-5x+6=0$

1. 2

Q.

The product of the ages of two sisters is $104$. The difference between their ages is $5$ years. Find their ages.

Q.

What are the solutions of the equation $x^4-5x^2-14=0$? Use factoring to solve.

Q. The roots of the quadratic equation $(x+\beta )(x-\alpha )=0$ are:

1. $\beta ,-\alpha$
2. $\alpha ,-\beta$
3. $-\alpha ,-\beta$
4. $\alpha ,\beta$

Q.

The longest rod that can be placed flat on the bottom of a box is $45\text{cm}$. The box is $9\text{cm}$ longer than it is wide. Find the length and breadth of the box.

Q. If one root of the equation $2x^2+ax-48=0$ is 3, then the value of a is ___________.

1. 10

Q. Find the sum of two consecutive positive integers, such that the sum of their squares is 365.
[3 Marks]

Q.

Out of herd of camels, $\frac{7}{2}$ times the square root of the total are grazing in the field. Two remaining are sitting in the field. Find the total number of camels.

Q.

$\text{Factorise the following by splitting the}\text{middle term and find the roots.}2x^2+x–6=0$

1. $x=2orx=-\frac{3}{2}$

2. $x=2orx=\frac{3}{2}$

3. $x=-2orx=\frac{3}{2}$

4. $x=-2orx=-\frac{3}{2}$

Q. Solve for $x$ if $4(2x+3{)}^2-(2x+3)-14=0$.

1. $x=-\frac{1}{2},\frac{19}{8}$
2. $x=\frac{1}{2},\frac{19}{8}$
3. $x=-\frac{1}{2},-\frac{19}{8}$
4. $x=\frac{1}{2},-\frac{19}{8}$

Q.

The hypotenuse of a right angled triangle is $6$ meters more than twice the shortest side. If the third side is $2$ meters less than the hypotenuse. find the sides of the triangle.

Q. Solve the following pair of linear equations by the elimination method and find the value of $p+q$ rounded off to the nearest integer.

Q.

The length of a hypotenuse of a right triangle exceeds the length of its base by $2\text{cm}$ and exceeds twice the length of the altitude by $1\text{cm}$. Find the length of each side of the triangle.

Q. By selling an item for ₹24, the shopkeeper earns the percent profit equal to cost price of the item. Find the cost price of the item.

1. 120
2. -120
3. -20
4. 20

Q.

Factorize $\sqrt{3}$ $x^2$ + 5x + 2$\sqrt{3}$ = 0 by splitting the middle term and find the value of x.

1. $x=\frac{-3}{\sqrt{2}},x=\sqrt{3}$

2. $x=\frac{2}{\sqrt{3}},x=\sqrt{3}$

3. $x=\frac{\sqrt{2}}{\sqrt{3}},x=-\sqrt{2}$

4. $x=\frac{-2}{\sqrt{3}},x=-\sqrt{3}$

Q. In a two digit number, the digit in one's place is 2 more than the digit in ten's place. Find the number if the product of digits is 48.

1. 46
2. 68
3. 86
4. 64

Q. Nikhil expressed the quadratic equation $x^2+10x+16=0$ in the form $x^2+px+qx+16=0$ such that the product of p and q is 16. Choose the correct values of p and q.

1. p = 2, q = 8
2. p = 2, q = 5
3. p = 1, q = 16
4. p = -2, q= -8

Q.

Solve following equation for $x$:
$x^2-3x-10=0$

1. -2 and -6

2. 10 and 5

3. 5 and -2

4. 7 and 6

Q.

Solve for x if $4(2x+3{)}^2-(2x+3)-14=0$.

1. $x=-\frac{1}{2},-\frac{19}{8}$

2. $x=\frac{1}{2},\frac{19}{8}$

3. $x=\frac{1}{2},-\frac{19}{8}$

4. $x=-\frac{1}{2},\frac{19}{8}$

Q.

By selling a table for ₹24, a trader loses as much percent as the cost price of the table. Calculate the cost price.

1. ₹60

2. ₹30

3. ₹40

4. ₹20

Q.

Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by $60$. Assume the middle number to be $x$ and form a quadratic equation satisfying the above statement. Hence, find the three numbers.