Solving a Quadratic Equation by Completing the Square

Q.

What number should be added to $x^2+6x$ to make it a perfect square?

1. 36

2. 18

3. 9

4. 72

Q.

If $x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$ then $x^2$ = ___

1. $161+72\sqrt{5}$

2. $9-4\sqrt{2}$

3. 161

4. $161-72\sqrt{5}$

Q.

Find the roots of the following quadratic equation if they exist by the method of completing the square : $2x^2+x-4=0$.

Q.

Solve the following equation using completing the square method.

$2x^2-x+\frac{1}{8}=0$

1. $x=-\frac{1}{4}$

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

3. $x=\frac{3}{4},\frac{3}{4}$

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

Q. Question 3 (i)
Find the roots of the following equations:
(i) $x-\frac{1}{x}=3,x\ne 0$

Q. Find the roots of the equation $5x^2–6x–2=0$ by the method of completing the square.

Q.

How do you evaluate $-7^2$?

Q.

Using the method of completion of squares find one of the roots of the equation $2x^2-7x+3=0$. Also, find the equation obtained after completion of the square.

1. 13, $(x-\frac{7}{2}{)}^2-\frac{25}{16}=0$

2. 3, $(x-\frac{7}{4}{)}^2-\frac{25}{16}=0$

3. 6, $(x-\frac{7}{4}{)}^2-\frac{25}{16}=0$

4. 3, $(x-\frac{7}{2}{)}^2-\frac{25}{16}=0$

Q. Solve the following equation by using the method of completing the square:
$\frac{2}{x^2}-\frac{5}{x}+2=0$

Q.

Question 7
Which constant must be added and subtracted to solve the quadratic equation $9x^2+\frac{3}{4}x-\sqrt{2}=0$ by the method of completing the square?
(A) $\frac{1}{8}$
(B) $\frac{1}{64}$
(C) $\frac{1}{4}$
(D) $\frac{9}{64}$

Q. Question 1(iii)
Find the roots of the following quadratic equations by factorisation:
(iii) $\sqrt{2}{x}^2+7x+5\sqrt{2}=0$

Q. $\text{Which of the following operations can}$
$\text{convert the expression}(x^2+\frac{8x}{\sqrt{3}})$
$\text{into a perfect square?}$

1. $\text{Addition of}{\left(\frac{8}{\sqrt{3}}\right)}^2.$
2. $\text{Addition of}{\left(\frac{4}{\sqrt{3}}\right)}^2.$
3. $\text{Addition of}{\left(\frac{8}{3}x\right)}^2.$
4. $\text{Subtraction of}\frac{8x}{\sqrt{3}}$

Q.

Find the roots of the equation $5x^2–6x–2=0$ by completing the square method.

1. $\frac{5\pm \sqrt{19}}{3}$

2. $\frac{3\pm \sqrt{19}}{5}$​

3. $5$

4. $3$

Q.

There are multiple chickens and rabbits in a cage.

There are $72$ heads and $200$ feet inside the cage.

How many chickens and rabbits are in there?

Q.

Form the pair of linear equations for the following problem and find their solution by substitution method:

The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of $10\mathrm{km}$, the charge paid is $\mathrm{Rs}.105$ and for a journey of $15\mathrm{km}$, the charge paid is $\mathrm{Rs}.155$. What are the fixed charges and the charge per km? How much does a person have to pay for traveling a distance of $25\mathrm{km}$?

Q. Solve the following equation by using the method of completing the square:
$7x^2+3x-4=0$

Q.

What is the converted form of the equation 5$x^2$ – 6$x$ – 2 = 0 after completing the square by the method of completion of the square?

1. (5x - 3)² + 19 = 0

2. (5x - 3)² – 19 = 0

3. (5x+3)² + 19 = 0

4. (5x + 3)² – 19 = 0

Q. Find the roots of the equation $x^2–14x+17=0$ by the method of completing the square.

1. $x=7\pm \sqrt{52}$
2. $x=11\pm \sqrt{32}$
3. $x=7\pm \sqrt{32}$
4. $x=10\pm \sqrt{32}$

Q.

Solve the equation 4$x^2$+ 7x = 1 by the method of completing the square. In one of the options, one of the roots and the converted form of the equation, after completing the square is given. Select this option.

1. x= (7+√65)/8 (2x+7/4)² – 65/4 = 0

2. x= (-7+√65)/8 (2x+7/4)² – 65/16 = 0

3. x= (-7-√65)/8 (x-7/2)² – 45/4 = 0

4. x= (3-√19)/2 (5x+3)² – 19 = 0

Q. Match the expressions on the left with the number to be added to it to make the resultant a perfect square.

1. 81
2. 4
3. 121

Q.

Solve the equation 5$x^2$– 6$x$– 2 = 0 by the method of completing the square. Find the positive root.

1. $x$= $\frac{3+\sqrt{19}}{5}$

2. $x$= $\frac{3+\sqrt{17}}{5}$

3. $x$= $\frac{3-\sqrt{19}}{5}$

4. $x$= $\frac{3+\sqrt{17}}{5}$

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

Q. Divide $(5x^2-25x+20)$ by $(x-1)$.

Q. There are three figures of areas $x^2,12x$ and $z$ in the below figure. Figures A and C are squares while figure B is a rectangle, with length of one side as $x.$ What should be the value of $z$ so that all the 3 figures can be arranged to form a square of side (x +6)?

1. 16
2. 36
3. 18
4. 24

Q.

Find the roots of the equation $5x^2–6x–2=0$ by the method of completing the square.

1. $\frac{5\pm \sqrt{19}}{3}$

2. $\frac{3\pm \sqrt{19}}{5}$​

3. $5$

4. $3$

Q.

What is the finalises form of the equation 5$x^2$ – 6$x$ – 2 = 0 by the method of completion of squares?

1. (5x - 3)² – 19 = 0

2. (5x - 3)² + 19 = 0

3. (5x + 3)² – 19 = 0

4. (5x+3)² + 19 = 0

Q. Formulate the following problems as a pair of equations, and hence find their solutions:
(i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

Q. Solve the following equation by using the method of completing the square:
$7x^2+3x-4=0$

Q. Solve the equation 5$x^2$– 6$x$– 2 = 0 by the method of completing the square. Find the positive root.

Q.

The product of 2 consecutive natural numbers is 56. Solve this problem using completing the square method of quadratic equation. Write the equation obtained after completing the square and also write the smaller of the 2 numbers.

1. 7, $(x+\frac{1}{2}{)}^2-\frac{225}{4}=0$

2. 8, $(x+\frac{1}{2}{)}^2–\frac{225}{4}=0$

3. 7, $(x-\frac{1}{2}{)}^2-\frac{225}{4}$ = 0

4. 8, $(x-\frac{1}{2}{)}^2-\frac{225}{4}=0$