Square of sum - Examples, Exercises and Solutions

To solve this problem, we will use the formula for the square of a sum, which is $(a+b)^2 = a^2 + 2ab + b^2$.

Here, we identify $a = x$ and $b = y$. Thus, our expression $(x+y)^2$ can be expanded as follows:

$(x+y)^2 = x^2 + 2xy + y^2$

Now we will match this expanded form with the given choices.

• Choice 1: $x^2 + y^2$ - This is missing the middle term $2xy$.
• Choice 2: $y^2 + x^2 + 2xy$ - This matches the expanded expression.
• Choice 3: $x^2 + xy + y^2$ - The term $xy$ doesn't match the required $2xy$.
• Choice 4: $x^2 - 2xy + y^2$ - The middle term has an incorrect sign.

Therefore, the expression that is equivalent to $(x+y)^2$ is choice 2: $y^2 + x^2 + 2xy$.

We use the abbreviated multiplication formula:

$x^2+2\times x\times3+3^2=$

$x^2+6x+9$

In this task, we are asked to simplify the formula using the abbreviated multiplication formulas.

Let's take a look at the formulas:

$(x-y)^2=x^2-2xy+y^2$

 $(x+y)^2=x^2+2xy+y^2$

$(x+y)\times(x-y)=x^2-y^2$

Taking into account that in the given exercise there is only addition operation, the appropriate formula is the second one:

Now let us consider, what number when multiplied by itself will equal 4 and what number when multiplied by itself will equal 25?

The answers are respectively 2 and 5:

We insert these into the formula:

$(2x+5)^2=$

$(2x+5)(2x+5)=$

$2x\times2x+2x\times5+2x\times5+5\times5=$

$4x^2+20x+25$

That means our solution is correct.

To solve the expression $(x^2 + 4)^2$, we will follow these steps:

• Step 1: Identify the expression as a binomial $(a + b)$, where $a = x^2$ and $b = 4$.
• Step 2: Apply the binomial square formula: $(a + b)^2 = a^2 + 2ab + b^2$.
• Step 3: Substitute $a$ and $b$ into the formula.
• Step 4: Calculate each term in the formula.
• Step 5: Simplify to arrive at the final expanded form.

Let's execute these steps:
Step 1: Identify $a = x^2$ and $b = 4$.
Step 2: Apply the formula $(a + b)^2 = a^2 + 2ab + b^2$.
Step 3: Substitute to get: $(x^2)^2 + 2(x^2)(4) + 4^2$.
Step 4: Calculate each term:
- $(x^2)^2 = x^4$,
- $2(x^2)(4) = 8x^2$,
- $4^2 = 16$.
Step 5: Combine the terms to get the expanded expression: $x^4 + 8x^2 + 16$.

Therefore, the solution to the expression is $x^4 + 8x^2 + 16$.

To solve this problem, follow these steps:

• Step 1: Identify the terms: Here, $a = x$ and $b = x^2$.
• Step 2: Apply the square of a binomial formula: $(a + b)^2 = a^2 + 2ab + b^2$.
• Step 3: Substitute the terms into the formula and simplify the resulting expression.

Now, let's work through each step:

Step 1: We have $a = x$ and $b = x^2$.
Step 2: The formula gives us: $(a + b)^2 = a^2 + 2ab + b^2$

Step 3: Substituting $a = x$ and $b = x^2$ into the formula:

This simplifies to:

Therefore, the expanded form of the expression $(x + x^2)^2$ is $\mathbf{x^2 + 2x^3 + x^4}$.

This matches with choice 3 from the options provided.

To solve this problem, we'll use the formula for the square of a sum.

• Step 1: Identify our variables as $a = 7$ and $b = 8$.
• Step 2: Apply the formula $(a + b)^2 = a^2 + 2ab + b^2$.
• Step 3: Substitute into the formula:
  $(7 + 8)^2 = 7^2 + 2 \times 7 \times 8 + 8^2$.
• Step 4: Express the final expression in the most informative form.

Therefore, the expanded expression for $(7 + 8)^2$ is $7^2 + 2 \times 7 \times 8 + 8^2$.

Regarding the choices provided, the correct one is Option 3: $7^2 + 2 \times 7 \times 8 + 8^2$.

To solve this problem, we'll follow these steps:

• Identify the given expression $(a+b)^2$.
• Apply the formula for the square of a sum: $(x+y)^2 = x^2 + 2xy + y^2$.
• Substitute $x = a$ and $y = b$ into the formula and simplify.

Let's apply these steps to the expression $(a+b)^2$:
We start with the expression $(a+b)^2$. This means we are squaring the sum $a + b$.

According to the formula $(x+y)^2 = x^2 + 2xy + y^2$, we can substitute $x = a$ and $y = b$. Therefore, the expression becomes:

$(a+b)^2 = a^2 + 2ab + b^2$.

Therefore, the expanded form of the expression $(a+b)^2$ is $a^2 + 2ab + b^2$.

According to the shortened multiplication formula:

Since 7 and X appear twice, we raise both terms to the power:

$(7+x)^2$

Let's solve the equation. First, we'll simplify the algebraic expressions using the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$We'll then apply the formula we mentioned and expand the parentheses in the expression in the equation:

$(x+3)^2=x^2+9 \\ x^2+2\cdot x\cdot3+3^2=x^2+9\\ x^2+6x+9=x^2+9$We'll continue and combine like terms, by moving terms around. Later - we can notice that the squared term cancels out, therefore it's a first-degree equation, which we'll solve by isolating the variable term on one side and dividing both sides of the equation by its coefficient:

$x^2+6x+9=x^2+9 \\ 6x=0\hspace{8pt}\text{/}:6\\ \boxed{x=0}$Therefore, the correct answer is answer A.

Let's solve the equation $x^2 + 16x + 64 = 81$ step-by-step:

• Step 1: Recognize the Left-Hand Side as a Perfect Square
  Notice that the left-hand side, $x^2 + 16x + 64$, can be factored as $(x + 8)^2$ because $(x + 8)^2 = x^2 + 16x + 64$.
• Step 2: Set the Factored Expression Equal to 81
  We have $(x + 8)^2 = 81$.
• Step 3: Take the Square Root of Both Sides
  Taking the square root on both sides gives two possible equations: $x + 8 = 9$ and $x + 8 = -9$.
• Step 4: Solve Each Equation
  For $x + 8 = 9$:
  Subtract 8 from both sides to get $x = 1$.
  For $x + 8 = -9$:
  Subtract 8 from both sides to get $x = -17$.

Therefore, the solutions to the equation are $x = -17$ or $x = 1$.

We will first solve the exercise by opening the inner brackets:

(2[x+3])²

(2x+6)²

We will then use the shortcut multiplication formula:

(X+Y)²=X²+2XY+Y²

(2x+6)² = 2x² + 2x*6*2 + 6² = 2x+24x+36

To solve this problem, let's start by expanding the expression on the right side of the equation using the formula for the square of a sum.

Step 1: Expand $(x + 3)^2$:

• Using the formula $(a + b)^2 = a^2 + 2ab + b^2$, we substitute $a = x$ and $b = 3$.
• We have: $(x + 3)^2 = x^2 + 2 \times x \times 3 + 3^2$.
• Simplifying, we get: $x^2 + 6x + 9$.

Step 2: Compare with the given expression:

• The left side of the equation is $x^2 + \text{?} + 9$.
• From our expansion, we see $x^2 + 6x + 9$ matches the structure $x^2 + \text{?} + 9$.
• This implies the missing term $\text{?}$ is $6x$.

Therefore, the missing term in the expression $x^2 + \text{?} + 9 = (x + 3)^2$ is $6x$.

To solve this problem, we'll use the formula for the square of a sum, $(a+b)^2 = a^2 + 2ab + b^2$. This will help us determine the missing components in the expression.

The expression given is $(x+?)^2 = x^2 + ? + 25$.

First, match it to the expanded form:

$(x+b)^2 = x^2 + 2bx + b^2$.

According to the problem, the expanded form is $x^2 + ? + 25$. This means $b^2 = 25$, so:
$b^2 = 25$
Therefore, $b = 5$ or $b = -5$. For simplicity, let's choose $b = 5$.

Now, calculate $2bx$:
$2bx = 2 \times 5 \times x = 10x$

Substituting $b$ into the equation, we have:

$(x+5)^2 = x^2 + 10x + 25$.

Thus, the missing numbers are $5$ and $10x$.

Therefore, the solution to the problem is $5, 10x$.

To solve the problem $(3x + 4)^2$, we will use the formula for the square of a binomial:

$(a + b)^2 = a^2 + 2ab + b^2$

• Identify $a$ and $b$: Here, $a = 3x$ and $b = 4$.
• Apply the formula:

1. Calculate $a^2$ which is $(3x)^2 = 9x^2$.

2. Calculate $2ab$ which is $2 \times (3x) \times 4 = 24x$.

3. Calculate $b^2$ which is $4^2 = 16$.

Combine the results:

• $a^2 + 2ab + b^2 = 9x^2 + 24x + 16$.

Therefore, the expanded form of $(3x + 4)^2$ is $9x^2 + 24x + 16$.

The correct answer choice is: $9x^2 + 24x + 16$.

To solve the quadratic equation $4x^2 = 12x - 9$, we begin by rewriting it in standard quadratic form:

$4x^2 - 12x + 9 = 0$

Here, we compare to the general form $ax^2 + bx + c = 0$ and identify:

• $a = 4$
• $b = -12$
• $c = 9$

We will now use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute in the values for $a$, $b$, and $c$:

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Simplify:

$x = \frac{12 \pm \sqrt{144 - 144}}{8}$

$x = \frac{12 \pm \sqrt{0}}{8}$

$x = \frac{12 \pm 0}{8}$

This simplifies further to:

$x = \frac{12}{8} = \frac{3}{2}$

Therefore, the solution to the equation $4x^2 = 12x - 9$ is $x = \frac{3}{2}$.