Difference of squares: Factoring trinomials using decomposition

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

• Step 1: Simplify the quadratic expression using the difference of squares formula.
• Step 2: Combine like terms and bring all terms to one side to form a standard quadratic equation.
• Step 3: Solve the quadratic equation using factoring or the quadratic formula.

Let's work through each step:
Step 1: The expression $(x-8)(x+8)$ can be expanded using the difference of squares formula: $(x-8)(x+8) = x^2 - 64.$

Substituting this into the equation: $x^2 - 64 + 3x + 17 = -49.$

Step 2: Combine like terms: $x^2 + 3x + 17 - 64 = -49.$

Simplifying further: $x^2 + 3x - 47 = -49.$

Add 49 to both sides to form the standard quadratic equation: $x^2 + 3x + 2 = 0.$

Step 3: Now, solve the quadratic equation. Notice that we can factor it: $x^2 + 3x + 2 = (x + 1)(x + 2) = 0.$

This gives us two possible solutions: $x + 1 = 0 \quad \Rightarrow \quad x = -1,$ $x + 2 = 0 \quad \Rightarrow \quad x = -2.$

Therefore, the solutions to the problem are $x = -1, -2$.

To solve the given equation $(x-4)(x+4) - 5x = -8 - x^2 - 25x - 38 + 4x$, we'll follow these steps:

Step 1: Expand the expression on the left side using the difference of squares formula:
$(x-4)(x+4) = x^2 - 16$.
Substituting, we get:
$x^2 - 16 - 5x$.

Step 2: Simplify the right-hand side expression:
Combine like terms: $-x^2 - 25x + 4x - 8 - 38$ becomes
$-x^2 - 21x - 46$.

We now have the equation:
$x^2 - 16 - 5x = -x^2 - 21x - 46$.

Step 3: Bring all terms to one side to equate to zero:
Add $x^2$ and add $21x$ to both sides:
$x^2 + x^2 - 5x + 21x - 16 + 46 = 0$.
This simplifies to:
$2x^2 + 16x + 30 = 0$.

Step 4: Simplify further by factoring or using the quadratic formula. Factor out the common term:
$x^2 + 8x + 15 = 0$.
This factors to:
$(x+5)(x+3) = 0$.

Setting each factor to zero gives:
$x+5 = 0$ or $x+3 = 0$.
Thus, $x = -5$ or $x = -3$.

Therefore, the solution to the equation is $x = -5, -3$.

To solve this quadratic equation, follow these steps:

• Step 1: Expand both sides completely.
• Step 2: Simplify both sides to form a quadratic equation.
• Step 3: Solve the quadratic equation using appropriate methods.

Now, let's work through the steps:

Step 1: Expand the expressions.

The left side is $x^2 - 13x + 3 + (x-3)(x+3)$. Using the difference of squares formula, expand $(x-3)(x+3)$ as:

Substituting back, the left side becomes:

The right side is $(x-6)(x+6)$. Expand using the difference of squares:

Step 2: Simplify both sides.

Combine terms on the left side:

The right side remains:

The equation becomes:

Subtract $x^2$ from both sides to simplify further:

Step 3: Solve for $x$.

Move -36 to the other side to form a standard quadratic equation:

Factor the quadratic:

Setting each factor to zero gives:

These lead to the solutions:

Thus, the solutions to the equation are $x = 3$ and $x = 10$.

Let's begin by focusing on the right side of the equation:

$(2x-1)(2x+1)$

We must first open the parentheses whilst multiplying all the terms as needed:

$2x\cdot2x+2x\cdot1+-1\cdot2x+-1\cdot1$

$4x^2+2x-2x-1$
$4x^2-1$

Let's now return to the original equation, and move all terms to the same side.

$5X^2+7x+9=4X^2-1$
$5X^2-4X^2+7x+9+1=0$
$X^2+7X+10=0$

We are left with a simple quadratic equation, which can be solved using any method we desire (factoring or the quadratic formula).

Therefore the final solution is:

$X= -2,-5$

To solve the problem, we'll start by simplifying the right side of the equation.

1. Begin by expanding the difference of squares on the right side:
$(x-6)(x+6) = x^2 - 6^2 = x^2 - 36$.

2. Substitute back into the equation:
$x^2 + 2x - 24 = -13 + (x^2 - 36) + 7x$.

3. Simplify the right side:
Combine like terms:
$-13 + x^2 - 36 + 7x = x^2 + 7x - 49$.

4. Now the equation is:
$x^2 + 2x - 24 = x^2 + 7x - 49$.

5. Subtract $x^2$ from both sides to eliminate $x^2$:
$2x - 24 = 7x - 49$.

6. Move all terms involving $x$ to one side and constants to the other side:
Subtract $7x$ from both sides:
$2x - 7x = -49 + 24$.
Simplify to:
$-5x = -25$.

7. Solve for $x$ by dividing both sides by $-5$:
$x = \frac{-25}{-5} = 5$.

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