Square of sum: System of equations with no solution

We use the abbreviated multiplication formula:

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

$x^2+6x+9$

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.

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

In order to solve the exercise, remember the abbreviated multiplication formulas:

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

Let's start by using the property in both cases:

$(x+3)^2=x^2+6x+9$

$(x+2)^2=x^2+4x+4$

We then reinsert them back into the formula as follows:

$2(x^2+6x+9)+3(x^2+4x+4)=$

$2x^2+12x+18+3x^2+12x+12=$

$5x^2+24x+30$

To solve this problem, we'll simplify the expression $(x+y+1)^2$ by recognizing it as a square of a sum involving three terms:

• Step 1: Use the formula $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$.
• Step 2: Identify $a = x$, $b = y$, and $c = 1$.
• Step 3: Substitute these values into the formula.
• Step 4: Calculate each square and product term.
• Step 5: Simplify the expression by combining all computed terms.

Now, let's work through the steps:

We start with the formula: $(x+y+1)^2 = x^2 + y^2 + 1^2 + 2xy + 2 \cdot x \cdot 1 + 2 \cdot y \cdot 1$

Calculate each component:

• $x^2$ remains $x^2$.
• $y^2$ remains $y^2$.
• $1^2$ results in $1$.
• The term $2xy$ results from the cross-product of $x$ and $y$.
• The term $2 \cdot x \cdot 1$ simplifies to $2x$.
• The term $2 \cdot y \cdot 1$ simplifies to $2y$.

Combine these elements to form the simplified expression:

$x^2 + y^2 + 1 + 2xy + 2x + 2y$

Thus, the simplified expression for $(x+y+1)^2$ is:

$x^2 + 2x + y^2 + 2y + 2xy + 1$.

This corresponds to choice number 4 in the provided options.

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

• Step 1: Expand $(x+5)^2$ using the formula for the square of a sum.
• Step 2: Compare the expanded expression with $x^2 + 9x + 24$.
• Step 3: Determine what additional expression is needed to make the two expressions equal.

Let's go through these steps in detail:
Step 1: First, we expand $(x+5)^2$ using the formula $(a+b)^2 = a^2 + 2ab + b^2$:
$(x+5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2 = x^2 + 10x + 25$.

Step 2: Now, compare the expanded expression $x^2 + 10x + 25$ with the given $x^2 + 9x + 24$.
We notice that the linear term 10x needs to be replaced or adjusted with 9x, and the constant 25 with 24.

Step 3: To make the expressions equal, find the difference in linear and constant terms:
$10x + 25$ must equal $9x + 24 + k$ (where $k$ is what we need to add):
Equating them, we get $10x + 25 = 9x + 24 + k$.
Solve for $k$:
$10x + 25 - 9x - 24 = k$.
$x + 1 = k$.

Therefore, the expression that should be added is $x+1$.