Solve the Quadratic: Finding x in 4x² = 12x - 9

Q: Why did the discriminant equal zero in this problem?

When the discriminant $ b^2 - 4ac = 0 $, it means the quadratic has exactly one solution (a repeated root). This happens when the quadratic is a perfect square trinomial like $ (2x - 3)^2 = 0 $.

Q: Could I have factored this instead of using the quadratic formula?

Yes! The equation $ 4x^2 - 12x + 9 = 0 $ factors as $ (2x - 3)^2 = 0 $. Since it's a perfect square, both methods give $ x = \frac{3}{2} $.

Q: How do I recognize a perfect square trinomial?

Look for the pattern $ a^2 - 2ab + b^2 $ or check if the discriminant equals zero. Here: $ (2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9 $.

Q: What if I made an arithmetic error in the discriminant?

Double-check each step: $ b^2 = (-12)^2 = 144 $ and $ 4ac = 4(4)(9) = 144 $. So $ 144 - 144 = 0 $. Always verify your arithmetic!

Q: Why is there only one answer choice that's correct?

Perfect square trinomials have one repeated solution, not two different solutions. The answer $ x = \frac{3}{2} $ occurs twice mathematically, but we write it once.

Quadratic Equations with Perfect Square Trinomials

$4x^2=12x-9$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Arrange the equation so that one side equals 0

00:14 Break down 4 into 2 squared

00:21 Break down 9 into 3 squared

00:26 Break down 12 into factors 2,2 and 3

00:33 Use the shortened multiplication formulas to find the brackets

00:41 Isolate X

00:55 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

$4x^2=12x-9$

Step-by-step solution

To solve the quadratic equation $4x^2 = 12x - 9$ , we begin by rewriting it in standard quadratic form:

$4x^2 - 12x + 9 = 0$

Here, we compare to the general form $ax^2 + bx + c = 0$ and identify:

$a = 4$
$b = -12$
$c = 9$

We will now use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute in the values for $a$ , $b$ , and $c$ :

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Simplify:

$x = \frac{12 \pm \sqrt{144 - 144}}{8}$

$x = \frac{12 \pm \sqrt{0}}{8}$

$x = \frac{12 \pm 0}{8}$

This simplifies further to:

$x = \frac{12}{8} = \frac{3}{2}$

Therefore, the solution to the equation $4x^2 = 12x - 9$ is $x = \frac{3}{2}$ .

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Standard Form: Move all terms to one side before solving
Discriminant Check: When b² - 4ac = 0, expect exactly one solution
Verify: Substitute $x = \frac{3}{2}$ back: 4(9/4) = 12(3/2) - 9 = 9 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to move all terms to one side
Don't try to solve 4x² = 12x - 9 directly without rearranging = wrong setup! This prevents you from identifying coefficients correctly for the quadratic formula. Always rewrite in standard form ax² + bx + c = 0 first.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

Why did the discriminant equal zero in this problem?

When the discriminant $b^2 - 4ac = 0$ , it means the quadratic has exactly one solution (a repeated root). This happens when the quadratic is a perfect square trinomial like $(2x - 3)^2 = 0$ .

Could I have factored this instead of using the quadratic formula?

Yes! The equation $4x^2 - 12x + 9 = 0$ factors as $(2x - 3)^2 = 0$ . Since it's a perfect square, both methods give $x = \frac{3}{2}$ .

How do I recognize a perfect square trinomial?

Look for the pattern $a^2 - 2ab + b^2$ or check if the discriminant equals zero. Here: $(2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9$ .

What if I made an arithmetic error in the discriminant?

Double-check each step: $b^2 = (-12)^2 = 144$ and $4ac = 4(4)(9) = 144$ . So $144 - 144 = 0$ . Always verify your arithmetic!

Why is there only one answer choice that's correct?

Perfect square trinomials have one repeated solution, not two different solutions. The answer $x = \frac{3}{2}$ occurs twice mathematically, but we write it once.

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Solve the Quadratic: Finding x in 4x² = 12x - 9

Quadratic Equations with Perfect Square Trinomials

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why did the discriminant equal zero in this problem?

Could I have factored this instead of using the quadratic formula?

How do I recognize a perfect square trinomial?

What if I made an arithmetic error in the discriminant?

Why is there only one answer choice that's correct?

🌟 Unlock Your Math Potential

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