Solve the Algebraic Expression: (x-8)(x+8) + 3x + 17 = -49

Q: How do I recognize when to use the difference of squares formula?

Look for the pattern (a-b)(a+b) where you have the same terms but opposite signs. Like $ (x-8)(x+8) $ - notice the 8's have opposite signs!

Q: What if I forget the difference of squares formula?

You can still expand using FOIL, but it's much longer! $ (x-8)(x+8) = x² + 8x - 8x - 64 = x² - 64 $. The middle terms always cancel out.

Q: How do I factor x² + 3x + 2?

Look for two numbers that multiply to 2 and add to 3. That's 1 and 2! So $ x² + 3x + 2 = (x + 1)(x + 2) $.

Q: Can I check my answers x = -1 and x = -2?

Yes! Substitute back into the original equation. For $ x = -1 $: $ ((-1)-8)((-1)+8) + 3(-1) + 17 = (-9)(7) - 3 + 17 = -63 - 3 + 17 = -49 ✓ $

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

Solve the equation:

^{$(x-8)(x+8)+3x+17=-49$}

2

Step-by-step solution

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 $x = -1, -2$ .

3

Final Answer

x = -1, -2

Key Points to Remember

Essential concepts to master this topic

Rule: Difference of squares formula: $(a-b)(a+b) = a^2 - b^2$
Technique: $(x-8)(x+8) = x^2 - 64$ then combine like terms
Check: Substitute $x = -1$ : $1 - 3 + 17 = 15 \neq -49$ Wait, let me recalculate ✓

Common Mistakes

Avoid these frequent errors

Expanding (x-8)(x+8) by distributing instead of using difference of squares
Don't expand (x-8)(x+8) as x²+8x-8x-64 = tedious and error-prone! Students often make sign errors or forget terms. Always recognize the difference of squares pattern and use (a-b)(a+b) = a²-b² directly.

Practice Quiz

Test your knowledge with interactive questions

Solve:

$$ (2+x)(2-x)=0 $$

FAQ

Everything you need to know about this question

How do I recognize when to use the difference of squares formula?

+

Look for the pattern (a-b)(a+b) where you have the same terms but opposite signs. Like $(x-8)(x+8)$ - notice the 8's have opposite signs!

What if I forget the difference of squares formula?

+

You can still expand using FOIL, but it's much longer! $(x-8)(x+8) = x² + 8x - 8x - 64 = x² - 64$ . The middle terms always cancel out.

Why did we get x² + 3x + 2 = 0 at the end?

+

After expanding and combining like terms, we moved everything to one side: $x² + 3x - 47 + 49 = 0$ , which simplifies to $x² + 3x + 2 = 0$ .

How do I factor x² + 3x + 2?

+

Look for two numbers that multiply to 2 and add to 3. That's 1 and 2! So $x² + 3x + 2 = (x + 1)(x + 2)$ .

Can I check my answers x = -1 and x = -2?

+

Yes! Substitute back into the original equation. For $x = -1$ : $((-1)-8)((-1)+8) + 3(-1) + 17 = (-9)(7) - 3 + 17 = -63 - 3 + 17 = -49 ✓$

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Solve the Algebraic Expression: (x-8)(x+8) + 3x + 17 = -49

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