Solve the Equation: (2x²-32)/8 = (x+4)/2 Step-by-Step

Q: Why do we cross-multiply instead of finding LCD?

Cross-multiplication works perfectly when you have one fraction equals another fraction. It's faster than finding LCD since we multiply $ (2x^2-32) \times 2 = 8 \times (x+4) $ directly!

Q: How do I know which factoring method to use?

For $ x^2 - 2x - 24 = 0 $, look for two numbers that multiply to -24 and add to -2. That's -6 and +4, giving us $ (x-6)(x+4) = 0 $.

Q: What if I get two different answers?

Check both solutions! Quadratic equations can have 0, 1, or 2 valid solutions. Always substitute each answer back into the original equation to see which ones actually work.

Q: Why did we divide the whole equation by 4?

Dividing by the greatest common factor makes numbers smaller and easier to work with. $ 4x^2 - 8x - 96 = 0 $ becomes the simpler $ x^2 - 2x - 24 = 0 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Take out 2 from the parentheses

00:21 Divide 8 by 2

00:28 Use the shortened multiplication formulas

00:43 Convert from 16 to 4 squared

00:56 Again use the shortened multiplication formulas

01:05 Multiply by 4 to eliminate the fraction

01:29 Reduce what we can

01:55 Isolate X

02:05 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$\frac{2x^2-32}{8}=\frac{x+4}{2}$

2

Step-by-step solution

To solve the equation $\frac{2x^2-32}{8} = \frac{x+4}{2}$ , let's proceed through these steps:

Step 1: Simplify and eliminate fractions by cross-multiplying.
Step 2: Rearrange and simplify the resulting equation.
Step 3: Solve for the variable $x$ .

Now, let's work through each step:

Step 1: Cross-multiply to eliminate the fractions.
We perform cross multiplication as follows:

\left(2x^2 - 32\right) \times 2 = 8 \times (x + 4)

This gives us:

4x^2 - 64 = 8x + 32

Step 2: Rearrange and simplify the equation.
Move all terms to one side to set the equation to zero:

4x^2 - 8x - 96 = 0

Simplify by dividing the entire equation by 4:

x^2 - 2x - 24 = 0

Step 3: Solve the quadratic equation by factoring.
Factor the quadratic equation:

(x - 6)(x + 4) = 0

Set each factor to zero and solve for $x$ :

$x - 6 = 0 \Rightarrow x = 6$
$x + 4 = 0 \Rightarrow x = -4$

Considering the given multiple-choice answers, the correct solution is:

$x = 6$

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

3

Final Answer

6

Key Points to Remember

Essential concepts to master this topic

Cross-Multiply: Multiply numerator by opposite denominator to eliminate fractions
Simplify: Factor out common terms like dividing $4x^2 - 8x - 96 = 0$ by 4
Verify: Substitute x = 6 back: $\frac{40}{8} = \frac{10}{2}$ gives 5 = 5 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check both solutions in original equation
Don't assume both solutions from factoring are valid = wrong final answer! One solution might create division by zero or not satisfy the original equation. Always substitute both x = 6 and x = -4 back into the original fractional equation to verify which ones work.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why do we cross-multiply instead of finding LCD?

+

Cross-multiplication works perfectly when you have one fraction equals another fraction. It's faster than finding LCD since we multiply $(2x^2-32) \times 2 = 8 \times (x+4)$ directly!

How do I know which factoring method to use?

+

For $x^2 - 2x - 24 = 0$ , look for two numbers that multiply to -24 and add to -2. That's -6 and +4, giving us $(x-6)(x+4) = 0$ .

What if I get two different answers?

+

Check both solutions! Quadratic equations can have 0, 1, or 2 valid solutions. Always substitute each answer back into the original equation to see which ones actually work.

Why did we divide the whole equation by 4?

+

Dividing by the greatest common factor makes numbers smaller and easier to work with. $4x^2 - 8x - 96 = 0$ becomes the simpler $x^2 - 2x - 24 = 0$ .

Can I use the quadratic formula instead?

+

Absolutely! The quadratic formula works for any quadratic equation. However, factoring is often faster when the numbers work out nicely like they do here.

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Solve the Equation: (2x²-32)/8 = (x+4)/2 Step-by-Step

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