Square of sum: Solving the problem

Equations with Variables on One SideUsing short multiplication formulasUsing variablesEquations with denominatorsComplete the missing numbersIdentify the greater valueUsing fractionsUsing quadrilateralsData with powers and rootsMore than one factorizationSystem of equations with no solutionUsing multiple rulesEquations with variables on both sidesNumber of termsTransition between expressionsResulting in a quadratic equation
Question 1

\( (x+1)^2=x^2 \)

Question 2

\( (x+2)^2-12=x^2 \)

Question 3

Solve for x:

\( (x+3)^2=x^2+9 \)

Question 4

\( (x+1)^2=x^2+13 \)

Question 5

\( (x-1)^2-(x+2)^2=15 \)

Examples with solutions for Square of sum: Solving the problem

Exercise #1

(x+1)2=x2 (x+1)^2=x^2

Video Solution

Step-by-Step Solution

Let's examine the given equation:

(x+1)2=x2 (x+1)^2=x^2 First, let's simplify the equation, using the perfect square binomial formula:

(a±b)2=a2±2ab+b2 (a\pm b)^2=a^2\pm2ab+b^2 ,

We'll start by opening the parentheses on the left side using the perfect square formula and then move terms and combine like terms, in the final step we'll solve the simplified equation we get:

(x+1)2=x2↓x2+2⋅x⋅1+12=x2x2+2x+1=x22x=−1/:2x=−12 (x+1)^2=x^2 \\ \downarrow\\ x^2+2\cdot x\cdot1+1^2=x^2\\ x^2+2x+1= x^2\\ 2x=-1\hspace{6pt}\text{/}:2\\ \boxed{x=-\frac{1}{2}} Therefore, the correct answer is answer A.

Answer

x=−12 x=-\frac{1}{2}

Exercise #2

(x+2)2−12=x2 (x+2)^2-12=x^2

Video Solution

Step-by-Step Solution

Let's examine the given equation:

(x+2)2−12=x2 (x+2)^2-12=x^2 First, let's simplify the equation, for this we'll use the perfect square binomial formula:

(a±b)2=a2±2ab+b2 (a\pm b)^2=a^2\pm2ab+b^2 ,

We'll start by opening the parentheses on the left side using the perfect square formula and then move terms and combine like terms, in the final step we'll solve the simplified equation we get:

(x+2)2−12=x2↓x2+2⋅x⋅2+22−12=x2x2+4x+4−12=x24x=8/:4x=2 (x+2)^2-12=x^2 \\ \downarrow\\ x^2+2\cdot x\cdot2+2^2-12=x^2\\ x^2+4x+4-12= x^2\\ 4x=8\hspace{6pt}\text{/}:4\\ \boxed{x=2} Therefore, the correct answer is answer C.

Answer

x=2 x=2

Exercise #3

Solve for x:

(x+3)2=x2+9 (x+3)^2=x^2+9

Video Solution

Answer

x=0 x=0

Exercise #4

(x+1)2=x2+13 (x+1)^2=x^2+13

Video Solution

Answer

x=6 x=6

Exercise #5

(x−1)2−(x+2)2=15 (x-1)^2-(x+2)^2=15

Video Solution

Answer

x=−3 x=-3

Question 1

Solve for x:

\( (x+2)^2=x^2+12 \)

Question 2

What is the value of x?

\( (x+3)^2=x^2+15 \)

Question 3

\( x^2+10x=-25 \)

Question 4

\( 4x^2=12x-9 \)

Question 5

Solve for y:

\( y^2+4y+2=-2 \)

Exercise #6

Solve for x:

(x+2)2=x2+12 (x+2)^2=x^2+12

Video Solution

Answer

x=2 x=2

Exercise #7

What is the value of x?

(x+3)2=x2+15 (x+3)^2=x^2+15

Video Solution

Answer

x=1 x=1

Exercise #8

x2+10x=−25 x^2+10x=-25

Video Solution

Answer

x=−5 x=-5

Exercise #9

4x2=12x−9 4x^2=12x-9

Video Solution

Answer

x=32 x=\frac{3}{2}

Exercise #10

Solve for y:

y2+4y+2=−2 y^2+4y+2=-2

Video Solution

Answer

y=−2 y=-2

Question 1

\( \frac{\sqrt{2x^2+4xy+2y^2+(x+y)^2}}{(x+y)}= \)

Question 2

Simply the following expression:

\( (x+\sqrt{x})^2 \)

Question 3

Solve the following equation:

\( \frac{1}{(x+1)^2}+\frac{1}{x+1}=1 \)

Question 4

Solve for x:

\( x^2+32x=-256 \)

Question 5

Look at the following equation:

\( \frac{\sqrt{x}-\sqrt{x+1}}{x+1}=1 \)

This can also be written as:

\( x[A(x+B)-x^3]=0 \)

Calculate A and B.

Exercise #11

2x2+4xy+2y2+(x+y)2(x+y)= \frac{\sqrt{2x^2+4xy+2y^2+(x+y)^2}}{(x+y)}=

Video Solution

Answer

3 \sqrt{3}

Exercise #12

Simply the following expression:

(x+x)2 (x+\sqrt{x})^2

Video Solution

Answer

x[x+2x+1] x\lbrack x+2\sqrt{x}+1\rbrack

Exercise #13

Solve the following equation:

1(x+1)2+1x+1=1 \frac{1}{(x+1)^2}+\frac{1}{x+1}=1

Video Solution

Answer

−12[1±5] -\frac{1}{2}[1\pm\sqrt{5}\rbrack

Exercise #14

Solve for x:

x2+32x=−256 x^2+32x=-256

Video Solution

Answer

x=−16 x=-16

Exercise #15

Look at the following equation:

x−x+1x+1=1 \frac{\sqrt{x}-\sqrt{x+1}}{x+1}=1

This can also be written as:

x[A(x+B)−x3]=0 x[A(x+B)-x^3]=0

Calculate A and B.

Video Solution

Answer

B=1 , A=4

Question 1

\( \frac{(\frac{1}{x}+\frac{1}{2})^2}{(\frac{1}{x}+\frac{1}{3})^2}=\frac{81}{64} \)

Find X

Question 2

Solve the following equation:

\( \frac{x^3+1}{(x+1)^2}=x \)

Question 3

Solve the following system of equations:

\( \begin{cases} \sqrt{x}+\sqrt{y}=\sqrt{\sqrt{61}+6} \\ xy=9 \end{cases} \)

Question 4

Look at the following equation:

\( \frac{\sqrt{x}+\sqrt{x+1}}{x+1}=1 \)

The same equation can be presented as follows:

\( x[A(x+B)-x^3]=0 \)

Calculate A and B.

Exercise #16

(1x+12)2(1x+13)2=8164 \frac{(\frac{1}{x}+\frac{1}{2})^2}{(\frac{1}{x}+\frac{1}{3})^2}=\frac{81}{64}

Find X

Video Solution

Answer

x=1,−177 x=1,-\frac{17}{7}

Exercise #17

Solve the following equation:

x3+1(x+1)2=x \frac{x^3+1}{(x+1)^2}=x

Video Solution

Answer

x=12 x=\frac{1}{2}

Exercise #18

Solve the following system of equations:

{x+y=61+6xy=9 \begin{cases} \sqrt{x}+\sqrt{y}=\sqrt{\sqrt{61}+6} \\ xy=9 \end{cases}

Video Solution

Answer

x=612−2.5 x=\frac{\sqrt{61}}{2}-2.5

y=612+2.5 y=\frac{\sqrt{61}}{2}+2.5

or

x=612+2.5 x=\frac{\sqrt{61}}{2}+2.5

y=612−2.5 y=\frac{\sqrt{61}}{2}-2.5

Exercise #19

Look at the following equation:

x+x+1x+1=1 \frac{\sqrt{x}+\sqrt{x+1}}{x+1}=1

The same equation can be presented as follows:

x[A(x+B)−x3]=0 x[A(x+B)-x^3]=0

Calculate A and B.

Video Solution

Answer

B=1 , A=4