Simplify and Solve: Find x in (x-4)(x+4) - 5x = -8 - x² - 25x + 4x - 38

Q: Why does (x-4)(x+4) equal x² - 16 and not something else?

This uses the difference of squares pattern: (a-b)(a+b) = a² - b². Here, a = x and b = 4, so you get x² - 4² = x² - 16. The middle terms cancel out!

Q: How do I know when to move all terms to one side?

To solve any equation, you need it to equal zero. Move all terms to one side so you can factor or use the quadratic formula on the simplified form.

Q: Why do I get two answers for x?

Quadratic equations typically have two solutions because they involve x². When you factor to get (x+5)(x+3) = 0, each factor can equal zero independently.

Q: What if I can't factor the quadratic?

If factoring is difficult, use the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $ for equations in the form ax² + bx + c = 0.

Q: How do I check if both x = -5 and x = -3 are correct?

Substitute each value into the original equation separately. Both should make the left side equal the right side. This confirms your solutions are correct!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Let's use the shortened multiplication formulas

00:29 Calculate 4 squared

00:34 Group terms

00:50 Arrange the equation so that the right side equals 0

01:07 Group terms

01:27 Divide by 2

01:35 Use the shortened multiplication formulas to find two numbers

01:38 whose sum equals 8

01:42 and their product equals 15

01:48 These are the numbers

01:57 Find the two possible solutions that make the equation equal zero

02:01 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Resolve:

^{$(x-4)(x+4)-5x=-8-x^2-25x-38+4x$}

2

Step-by-step solution

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 $x = -5, -3$ .

3

Final Answer

x=-5,-3

Key Points to Remember

Essential concepts to master this topic

Expansion: Use difference of squares formula for (x-4)(x+4) = x² - 16
Technique: Combine like terms: -25x + 4x becomes -21x on right side
Check: Substitute x = -5: (-5+5)(-5+3) = 0 × (-2) = 0 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand (x-4)(x+4) correctly
Don't multiply to get x² - 16x + 16 = wrong expansion! This creates extra terms that don't belong. Always use difference of squares: (a-b)(a+b) = a² - b², so (x-4)(x+4) = x² - 16.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why does (x-4)(x+4) equal x² - 16 and not something else?

+

This uses the difference of squares pattern: (a-b)(a+b) = a² - b². Here, a = x and b = 4, so you get x² - 4² = x² - 16. The middle terms cancel out!

How do I know when to move all terms to one side?

+

To solve any equation, you need it to equal zero. Move all terms to one side so you can factor or use the quadratic formula on the simplified form.

Why do I get two answers for x?

+

Quadratic equations typically have two solutions because they involve x². When you factor to get (x+5)(x+3) = 0, each factor can equal zero independently.

What if I can't factor the quadratic?

+

If factoring is difficult, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ for equations in the form ax² + bx + c = 0.

How do I check if both x = -5 and x = -3 are correct?

+

Substitute each value into the original equation separately. Both should make the left side equal the right side. This confirms your solutions are correct!

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Simplify and Solve: Find x in (x-4)(x+4) - 5x = -8 - x² - 25x + 4x - 38

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