Short Multiplication Formulas - 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:

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

$x^2+6x+9$

We use the abbreviated multiplication formula:

$(x-y)(x-y)=$

$x^2-xy-yx+y^2=$

$x^2-2xy+y^2$

To solve the problem, we need to expand the expression $(x-7)^2$ using the formula for the square of a difference.

The formula for the square of a difference is $(a-b)^2 = a^2 - 2ab + b^2$.

Let's apply this formula to our expression $(x-7)^2$:

• Identify $a = x$ and $b = 7$.
• Substitute these values into the formula: $(x-7)^2 = x^2 - 2(x)(7) + 7^2$.
• Calculate each term:
• $x^2$ remains as $x^2$.
• $-2(x)(7) = -14x$.
• $7^2 = 49$.

So, expanding the expression, we get $x^2 - 14x + 49$.

Thus, the expression that has the same value as $(x-7)^2$ is $x^2 - 14x + 49$.

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

• Step 1: Identify a and b in the expression $(x^2 - 6)^2$.
• Step 2: Apply the square of a difference formula.
• Step 3: Simplify the resulting expression.

Now, let's work through each step:
Step 1: The expression is $(x^2 - 6)^2$. Here, $a = x^2$ and $b = 6$.
Step 2: Apply the binomial formula: $(a - b)^2 = a^2 - 2ab + b^2$.
Step 3:
1. Calculate $a^2$:
$a^2 = (x^2)^2 = x^4$.
2. Calculate $2ab$:
$2ab = 2(x^2)(6) = 12x^2$.
3. Calculate $b^2$:
$b^2 = 6^2 = 36$.
4. Substitute these back into the formula:
$(x^2 - 6)^2 = x^4 - 12x^2 + 36$.

Therefore, the expanded expression is $x^4 - 12x^2 + 36$.

To solve the expression $(x-x^2)^2$, we will use the square of a binomial formula $(a-b)^2 = a^2 - 2ab + b^2$.

Let's identify $a$ and $b$ in our expression:

• Here, $a = x$ and $b = x^2$.

Applying the formula:

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

Calculating each part, we get:

• $(x)^2 = x^2$
• $-2(x)(x^2) = -2x^3$
• $(x^2)^2 = x^4$

Combining these results, the expression simplifies to:

$x^4 - 2x^3 + x^2$

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

We use the abbreviated multiplication formula:

$4-x^2=0$

We isolate the terms and extract the root:

$4=x^2$

$x=\sqrt{4}$

$x=\pm2$

To solve this problem, we will apply the square of a binomial formula.

The given expression is $(4b-3)(4b-3)$. We recognize this as the square of a binomial, which can be rewritten as $(4b-3)^2$. To expand this expression, we use the formula:

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

In our expression, $a = 4b$ and $b = 3$. Let's apply the formula:

• Calculate $a^2$:
  $a^2 = (4b)^2 = 16b^2$
• Calculate $-2ab$:
  $-2ab = -2(4b)(3) = -24b$
• Calculate $b^2$:
  $b^2 = (3)^2 = 9$

Putting it all together, we have:

$(4b-3)^2 = 16b^2 - 24b + 9$

Therefore, the exponential summation expression is $(4b-3)^2$, with the expanded form:

$16b^2 - 24b + 9$

This matches choice 3, confirming our solution.

To solve this problem, let's start by identifying the parts of the binomial:

• The expression $(3x-y)^2$ represents a binomial squared.
• We recognize it has the form $(a-b)^2$ where $a = 3x$ and $b = y$.
• Using the formula for the square of a difference: $(a-b)^2 = a^2 - 2ab + b^2$, we find the expanded form.

Let's apply the formula:

Step 1: Expand $(3x-y)^2$ using the formula:
$(3x-y)^2 = (3x)^2 - 2(3x)(y) + y^2$

Step 2: Calculate each part:
$(3x)^2 = 9x^2$
$-2(3x)(y) = -6xy$
$y^2$ stays as $y^2$

Step 3: Combine these results to get the addition form:
$9x^2 - 6xy + y^2$

The expression in multiplication form, as provided, is just repeating the factors:
$(3x-y)(3x-y)$

Therefore, the expression rewritten as addition is $9x^2 - 6xy + y^2$ and as multiplication $(3x-y)(3x-y)$.

To solve the problem, we will expand the expression $(a-4)(a-4)$ using the square of a difference formula.

This formula states: $(x-y)^2 = x^2 - 2xy + y^2$.
In our case, $x = a$ and $y = 4$, so we apply the formula:

• First term: $x^2 = a^2$
• Second term: $-2xy = -2 \cdot a \cdot 4 = -8a$
• Third term: $y^2 = 4^2 = 16$

Putting it all together, the expression becomes:
$a^2 - 8a + 16$.

After matching this result with the given choices, we find it corresponds to choice 4.

Therefore, the solution to the problem is $\mathbf{a^2 - 8a + 16}$.

To solve for $(7b - 3x)^2$ as a sum, we'll follow these steps:

• Step 1: Identify the given expression and apply the formula for the square of a difference:
  $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 7b$ and $b = 3x$.
• Step 2: Expand each term:
• $a^2 = (7b)^2 = 49b^2$
• $-2ab = -2 \times 7b \times 3x = -42bx$
• $b^2 = (3x)^2 = 9x^2$
• Step 3: Combine all terms to form the sum:
  $(7b - 3x)^2 = 49b^2 - 42bx + 9x^2$.

Therefore, the solution to the problem is $(7b - 3x)^2 = 49b^2 - 42bx + 9x^2$.

Hence, the correct answer choice is: $49b^2 - 42bx + 9x^2$

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 rewrite the expression $9x^2 - 12x + 4$ as a perfect square, follow these steps:

• Step 1: Compare the expression $9x^2 - 12x + 4$ with $(a - b)^2$.
• Step 2: Note that $(a - b)^2 = a^2 - 2ab + b^2$.
• Step 3: Identify matching terms: $a^2 = 9x^2$, $-2ab = -12x$, $b^2 = 4$.

Now, separate this into steps:
Step 1: Set $a^2 = 9x^2$, so $a = 3x$.
Step 2: Set $b^2 = 4$, so $b = 2$.
Step 3: Verify $-2ab = -12x$:
$-2 \times 3x \times 2 = -12x.$
This confirms our values of $a$ and $b$ are correct.

Thus, the expression $9x^2 - 12x + 4$ is equivalent to the square $(3x - 2)^2$.

The correct choice is: $(3x-2)^2$.