Solve the Equation: (2x)² - 3 = 6 Step-by-Step

Q: Why do we get two solutions for this equation?

Because we're solving a quadratic equation! When we factor $ 4x^2 - 9 = 0 $, we get $ (2x-3)(2x+3) = 0 $. For this to equal zero, either factor can be zero, giving us two solutions.

Q: Can I solve this without factoring?

Yes! You could also solve $ 4x^2 = 9 $ by dividing both sides by 4 to get $ x^2 = \frac{9}{4} $, then taking the square root: $ x = ±\frac{3}{2} $.

Q: What does the ± symbol mean?

The ± symbol means "plus or minus" - it's shorthand for writing both solutions. So $ x = ±\frac{3}{2} $ means $ x = +\frac{3}{2} $ or $ x = -\frac{3}{2} $.

Q: How do I recognize a difference of squares?

Look for the pattern $ a^2 - b^2 $! In our problem, $ 4x^2 - 9 $ becomes $ (2x)^2 - 3^2 $, which factors as $ (2x-3)(2x+3) $. Perfect squares like 4, 9, 16, 25 are key clues!

Q: Why don't I just solve (2x)² = 9 directly?

You absolutely can! After rearranging to get $ (2x)^2 = 9 $, take the square root: $ 2x = ±3 $, then divide by 2 to get $ x = ±\frac{3}{2} $. Both methods work!

Quadratic Equations with Perfect Square Factoring

$(2x)^2-3=6$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:05 Always solve parentheses first

00:10 When there's a power on a product of terms, each term is raised to that power

00:18 Arrange the equation so that 0 is on the right side

00:32 Take out 4 from the parentheses

00:42 Convert 9 to 3 squared and 4 to 2 squared

00:51 A squared fraction raises both numerator and denominator to the square

00:55 Use the shortened multiplication formulas

01:15 Find the two possible solutions

01:33 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(2x)^2-3=6$

Step-by-step solution

First we rearrange the equation and set it to 0

$4x^2-3-6=0$

$4x^2-9=0$

We then apply the shortcut multiplication formula:

$4(x^2-\frac{9}{4})=0$

$x^2-(\frac{3}{2})^2=0$

$(x-\frac{3}{2})(x+\frac{3}{2})=0$

$x=\pm\frac{3}{2}$

Final Answer

$±\frac{3}{2}$

Key Points to Remember

Essential concepts to master this topic

Expansion: $(2x)^2 = 4x^2$ before solving the equation
Technique: Factor $4x^2 - 9 = (2x)^2 - 3^2 = (2x-3)(2x+3)$
Check: Substitute $x = \frac{3}{2}$ : $(2 \cdot \frac{3}{2})^2 - 3 = 9 - 3 = 6$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand (2x)² correctly
Don't write (2x)² = 2x² instead of 4x² = wrong coefficient! This gives you x² - 9 = 0 instead of 4x² - 9 = 0, leading to incorrect solutions ±3. Always remember (2x)² = 2² × x² = 4x².

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 get two solutions for this equation?

Because we're solving a quadratic equation! When we factor $4x^2 - 9 = 0$ , we get $(2x-3)(2x+3) = 0$ . For this to equal zero, either factor can be zero, giving us two solutions.

Can I solve this without factoring?

Yes! You could also solve $4x^2 = 9$ by dividing both sides by 4 to get $x^2 = \frac{9}{4}$ , then taking the square root: $x = ±\frac{3}{2}$ .

What does the ± symbol mean?

The ± symbol means "plus or minus" - it's shorthand for writing both solutions. So $x = ±\frac{3}{2}$ means $x = +\frac{3}{2}$ or $x = -\frac{3}{2}$ .

How do I recognize a difference of squares?

Look for the pattern $a^2 - b^2$ ! In our problem, $4x^2 - 9$ becomes $(2x)^2 - 3^2$ , which factors as $(2x-3)(2x+3)$ . Perfect squares like 4, 9, 16, 25 are key clues!

Why don't I just solve (2x)² = 9 directly?

You absolutely can! After rearranging to get $(2x)^2 = 9$ , take the square root: $2x = ±3$ , then divide by 2 to get $x = ±\frac{3}{2}$ . Both methods work!

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