Square of Difference: Complete the missing numbers

To address this mathematical problem, we will apply the square of a binomial formula and solve for the missing term. Here's how:

• Step 1: Expand the right-hand side of the equation $(x-3)^2$.
• Step 2: Equate the expanded form to $x^2 + ? + 9$.
• Step 3: Solve for the missing term by comparing the coefficients.

Step 1: Expanding $(x-3)^2$ using the formula, we get:

$(x - 3)^2 = x^2 - 2 \cdot x \cdot 3 + 3^2$.

This simplifies to:

$x^2 - 6x + 9$.

Step 2: Equating this to the left-hand side:

$x^2 + ? + 9 = x^2 - 6x + 9$.

Step 3: Compare the terms:

The term that replaces "?" on the left-hand side must make the equation hold.

Setting corresponding terms equal, we find that:

$? = -6x$.

Therefore, the solution to the problem is $-6x$.

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

• Step 1: Identify the given form and match it with the square of a difference formula
• Step 2: Determine the values of 'a' and 'b' such that the expanded form matches both sides of the equation
• Step 3: Calculate the missing value in the expression

Now, let's work through each step:
Step 1: We are given the expression $(2x-?)^2=4x^2-12x+\text{?}$.
Step 2: Using the standard formula for a perfect square expansion:
$(2x-b)^2 = (2x)^2 - 2 \cdot 2x \cdot b + b^2 = 4x^2 - 4xb + b^2$.
By matching coefficients, in $4x^2 - 12x + \text{?}$, we see $4xb = 12x$. Thus, $b = 3$.
Step 3: Substitute $b = 3$ into $b^2$ to get the constant term: $b^2 = 3^2 = 9$.

Therefore, the solution to the problem is $3,\text{ }9$.

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

• Step 1: Recognize this as a binomial square problem.
• Step 2: Identify the expanded form of a binomial square.
• Step 3: Match the terms from both sides of the equation.

Now, let's work through each step:
Step 1: The given expression $(x-?)^2 = x^2 - ? + 25$ is the expansion of a binomial $(x-a)^2$.
Step 2: Recall that $(x-a)^2 = x^2 - 2ax + a^2$.
Step 3: Compare this equivalent form to $x^2 - ? + 25$:

- The term $x^2$ matches directly on both sides.
- The constant term $a^2 = 25$, so $a = 5$ because $5^2 = 25$.
- The middle term $-2ax$ is currently unspecified, but it provides the form needed to fill the blank with $-2ax$.

Matching the expanded form: - The term inside $(x-?)$ should be $5$ (because $a=5$).
- Therefore, the missing linear term can be $10x$ since $-2 \times 5 = -10$.

Therefore, the solution to the problem is $5,\text{ }10x$.

To solve this problem, let's identify the steps needed to determine the missing values.

• Step 1: Apply the perfect square formula.
  Write $(a \times x - b)^2 = a^2x^2 - 2abx + b^2$.
• Step 2: Compare with the given polynomial.
  Set the expression equal to the provided polynomial: $9x^2 - 24x + 16$.
• Step 3: Match coefficients.
  From $a^2x^2 = 9x^2$, we get $a^2 = 9$.
  From $-2abx = -24x$, we get $-2ab = -24$.
  From $b^2 = 16$, we get $b^2 = 16$.
• Step 4: Solve for $a$ and $b$.
  - Solve $a^2 = 9$, giving $a = 3$ or $a = -3$. Choose $a = 3$ for simplicity.
  - Solve $b^2 = 16$, giving $b = 4$ or $b = -4$. Choose $b = 4$ for simplicity.
  - Check $-2ab = -24$: $-2(3)(4) = -24$, which is correct.

Therefore, the missing values are $\boxed{3, 4}$.

Thus, the solution to the problem is: $3, 4$.

To solve this problem, let's consider the expansion of the expressions:

First, expand each square:

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

Expanding these expressions, we get:

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

Adding the two expanded forms gives:

$x^4 - 2x^2y + y^2 + y^4 - 2y^2x + x^2 = x^4 + y^4 + x^2 + y^2 - 2x^2y - 2y^2x$

Now rearrange the terms in a form similar to the given expression:

The goal is to arrange the polynomial as: $a(x^2 + 1) + 2xbcxy + c(y^2 + 1)$.

Since now we want: $(x^2 - y)^2 + (y^2 - x)^2 = a(x^2 + 1) + 2xy(- (x+y)) + c(y^2 + 1)$.

Create matching forms:

• The term for $a(x^2 + 1)$ will have $a = x^2$.
• The term for $b$ in $2xy(- (x+y))$ will have $b = -(x+y)$.
• The term for $c(y^2 + 1)$ will have $c = y^2 + 1$.

Therefore, the correct values to fill in are $(x^2,-(x+y),y^2+1)$.

Thus, our complete expression is:

$(x^2 - y)^2 + (y^2 - x)^2 = x^2(x^2+1) + 2xy(-(x+y)) + y^2(y^2+1)$

Hence, the answer is:

$x^2,-(x+y),y^2+1$

To solve this problem, we'll fill in the blanks in the expression $(?\times x-1)^2+(?\times x-2)^2=5x^2-8x+5$.

Let's expand the individual expressions:
1. $(a \cdot x - 1)^2 = a^2x^2 - 2ax + 1$
2. $(b \cdot x - 2)^2 = b^2x^2 - 4bx + 4$

Now, add these expansions together:
$(a^2x^2 - 2ax + 1) + (b^2x^2 - 4bx + 4)$
= $(a^2 + b^2)x^2 - (2a + 4b)x + 5$

We equate this to the given quadratic: $5x^2 - 8x + 5$.

Matching coefficients, we have:
$a^2 + b^2 = 5$
$2a + 4b = 8$

Now, solve these equations:

From the second equation: $2a + 4b = 8$, divide by 2:
$a + 2b = 4$

Substitute $a = 4 - 2b$ from this into $a^2 + b^2 = 5$:

$(4 - 2b)^2 + b^2 = 5$
$16 - 16b + 4b^2 + b^2 = 5$
$5b^2 - 16b + 16 = 5$
$5b^2 - 16b + 11 = 0$

Factoring the quadratic equation $5b^2 - 16b + 11 = 0$:
$(5b - 11)(b - 1) = 0$

This gives us $b = \frac{11}{5}$ or $b = 1$.

Using $b = 1$, substitute back to find $a$:
$a = 4 - 2(1) = 2$

Thus, the correct integers for the blanks are $a = 2$ and $b = 1$.

Therefore, the correct answer is $2, \text{ } 1$.

Let's solve the problem by following these detailed steps:

Step 1: Expand Both Sides

• Left Side: $(3x-a)^2 = 9x^2 - 6ax + a^2$
• $(4x-b)^2 = 16x^2 - 8bx + b^2$
• Right Side: $(x-c)^2 = x^2 - 2cx + c^2$
• $(2\sqrt{6}x-d)^2 = 24x^2 - 4\sqrt{6}xd + d^2$

Step 2: Form Complete Expanded Equations

Left Side: $9x^2 - 6ax + a^2 + 16x^2 - 8bx + b^2 = 25x^2 - (6a + 8b)x + (a^2 + b^2)$

Right Side: $x^2 - 2cx + c^2 + 24x^2 - 4\sqrt{6}xd + d^2 = 25x^2 - (2c + 4\sqrt{6}d)x + (c^2 + d^2)$

Step 3: Equate Coefficients

• $x^2$ terms: Coefficient is already checked as equal.
• $- (6a + 8b) = - (2c + 4\sqrt{6}d)$ implies $6a + 8b = 2c + 4\sqrt{6}d$ (1)
• Equate constant terms: $a^2 + b^2 = c^2 + d^2$ (2)

Step 4: Solve the System

• Choose condition $b > 0$ and analyze choice details:
• Using equations and choice alignment, correct viable numbers satisfy both conditions given choice and direct solve constraints.
• Solution: $a = -4\sqrt{6}, b = 3\sqrt{6}, c = -12, d = \sqrt{6}$

Therefore, the values are $a=-4\sqrt{6},b=3\sqrt{6},c=-12,d=\sqrt{6}$.