Extended Distributive Property: Solving an equation using the extended distributive law

To solve the given equation $(5-x)(6+2x)=-2+4x$, we'll apply the following steps to use the distributive property:

• Step 1: Expand the left side using the distributive property:
  $(5-x)(6+2x) = 5 \cdot 6 + 5 \cdot 2x - x \cdot 6 - x \cdot 2x$ $= 30 + 10x - 6x - 2x^2$ Finally, combine like terms:
  $= -2x^2 + 4x + 30$
• Step 2: Set the equation against the right side:
  $-2x^2 + 4x + 30 = -2 + 4x$
• Step 3: Move all terms to one side of the equation:
  $-2x^2 + 4x + 30 - 4x - (-2) = 0$ Simplify to:
  $-2x^2 + 32 = 0$
• Step 4: Solve the quadratic equation:
  Factor out the common term:
  $-2(x^2 - 16) = 0$
  $x^2 - 16 = 0$
• Step 5: Solve for $x$:
  $x^2 = 16$
  $x = \pm 4$

Therefore, the solution to the equation is $x = \pm 4$.

To solve the equation $(3x+4)(x+2) = 3x^2 + 2$, we start by expanding the left-hand side using the distributive property.

First, distribute each component of the first polynomial:

$(3x+4)(x+2) = 3x(x+2) + 4(x+2)$

Next, distribute inside each term:

$3x(x+2) = 3x \cdot x + 3x \cdot 2 = 3x^2 + 6x$$4(x+2) = 4 \cdot x + 4 \cdot 2 = 4x + 8$

Combining these, we have:

$3x^2 + 6x + 4x + 8 = 3x^2 + 10x + 8$

Set the expanded expression equal to the right side of the original equation:

$3x^2 + 10x + 8 = 3x^2 + 2$

To solve for $x$, subtract $3x^2$ from both sides:

$10x + 8 = 2$

Next, subtract 8 from both sides to isolate the term involving $x$:

$10x = 2 - 8$

$10x = -6$

Finally, divide both sides by 10:

$x = \frac{-6}{10}$$x = -0.6$

Therefore, the solution to the equation is $-0.6$.

The correct choice from the provided options is $\boxed{-0.6}$.

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

• Step 1: Expand the expression using the distributive property.
• Step 2: Compare the coefficients and constant terms to solve for a relationship between $a$ and $x$.

Now, let's work through each step:

Step 1:
The given equation is $(2x+a)(a-4) = 2ax + a^2 - 5$. First, expand the left-hand side:

$(2x + a)(a - 4)$

Using the distributive property:

• $2x \cdot a = 2ax$
• $2x \cdot -4 = -8x$
• $a \cdot a = a^2$
• $a \cdot -4 = -4a$

Combining these terms gives:

$2ax - 8x + a^2 - 4a$

Step 2:
Now, we set the expanded left-hand side equal to the right-hand side from the original equation:

$2ax - 8x + a^2 - 4a = 2ax + a^2 - 5$

Cancel the common terms on both sides:

• Subtract $2ax$ from both sides: $-8x - 4a = -5$
• Subtract $a^2$ from both sides: No change needed since both sides already equal.

The equation becomes:

$-8x - 4a = -5$

Solving for $a$:

Add $4a$ to both sides:

$-8x = 4a - 5$

Divide each term by 4 to solve for $a$:

$a = -2x + \frac{5}{4}$

Expressing in a simpler equivalent format, we have:

$a = -2x + 1\frac{1}{4}$

Therefore, we find the relationship between $a$ and $x$ to be $a = 1\frac{1}{4} - 2x$.

To solve the equation $(8x+9)(5-x) = 31x + 94$, we will use the distributive property. The steps are as follows:

• Step 1: Apply the distributive property to the left-hand side.
  We do this by multiplying each term in the first binomial by each term in the second one:
• $(8x + 9)(5 - x) = 8x \cdot 5 + 8x \cdot (-x) + 9 \cdot 5 + 9 \cdot (-x)$.
• Simplify the terms:
  $8x \cdot 5 = 40x$,
  $8x \cdot (-x) = -8x^2$,
  $9 \cdot 5 = 45$,
  $9 \cdot (-x) = -9x$.
• Combine all like terms:
  The left-hand side becomes $-8x^2 + 40x + 45 - 9x = -8x^2 + 31x + 45$.
• Step 2: Compare both sides of the equation:
  We have $-8x^2 + 31x + 45$ on the left-hand side and $31x + 94$ on the right.
• Step 3: Set the equation:
  $-8x^2 + 31x + 45 = 31x + 94$.
• Step 4: Subtract $31x$ from both sides:
  $-8x^2 + 45 = 94$.
• Step 5: Now subtract 45 from both sides:
  $-8x^2 = 49$.
• Step 6: Divide by $-8$:
  $x^2 = -\frac{49}{8}$.
• Step 7: Evaluate the Result:
  The result $x^2 = -\frac{49}{8}$ implies taking the square root of a negative number, which results in an imaginary number, thus indicating no real solution exists.

Therefore, the solution to the equation is: There is no solution to the equation.