Solve Quadratic Equation: 4x² + 24x + 36 = 0 Step-by-Step

Q: How do I recognize a perfect square trinomial?

Look for the pattern $ a^2 + 2ab + b^2 $! In $ 4x^2 + 24x + 36 $, we have (2x)² + 2(2x)(6) + 6², which forms the perfect square $ (2x + 6)^2 $.

Q: Why does this equation only have one solution?

When a quadratic factors as $ (expression)^2 = 0 $, it's called a repeated root. The same solution x = -3 appears twice, so technically there's only one unique answer.

Q: What if I can't see the perfect square pattern?

Start by factoring out common factors first! Here, factor out 4: $ 4(x^2 + 6x + 9) = 0 $. Then $ x^2 + 6x + 9 = (x + 3)^2 $ is easier to spot.

Q: Can I solve this using other methods?

Yes! You could use the quadratic formula or completing the square, but factoring perfect squares is the fastest method. All methods will give the same answer: x = -3.

Q: How do I verify my factoring is correct?

Always expand your factored form! Multiply out $ (2x + 6)^2 $ to get $ 4x^2 + 24x + 36 $. If it matches the original equation, your factoring is right!

Perfect Square Trinomials with Factoring

Solve the following equation:

$4x^2+24x+36=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Pay attention to the coefficients

00:10 Use the roots formula

00:24 Substitute appropriate values according to the given data and solve for X

00:43 Calculate the products and the square

01:06 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

Solve the following equation:

$4x^2+24x+36=0$

Step-by-step solution

To solve the quadratic equation $4x^2 + 24x + 36 = 0$ , we will simplify it by factoring:

First, notice that the given equation can be simplified as a perfect square:

Recognize that $4x^2$ , $24x$ , and $36$ can form a perfect square trinomial: $(2x + 6)^2$ .
Expand $(2x + 6)^2$ to verify it corresponds to the original equation:
$(2x + 6)^2 = (2x + 6)(2x + 6) = 4x^2 + 12x + 12x + 36 = 4x^2 + 24x + 36$ .

The equation has now been verified to be a perfect square: $\left(2x + 6\right)^2 = 0$ .

Set $2x + 6 = 0$ , and solve for $x$ :

Subtract 6 from both sides: $2x = -6$ .
Divide both sides by 2: $x = -3$ .

Thus, the solution to the quadratic equation is $\boxed{x = -3}$ .

Final Answer

$x=-3$

Key Points to Remember

Essential concepts to master this topic

Recognition: Identify perfect square patterns like $a^2 + 2ab + b^2$
Technique: Factor $4x^2 + 24x + 36$ as $(2x + 6)^2$
Check: Expand $(2x + 6)^2 = 4x^2 + 24x + 36$ to verify ✓

Common Mistakes

Avoid these frequent errors

Using quadratic formula on perfect squares
Don't apply the quadratic formula to $4x^2 + 24x + 36 = 0$ = complicated calculations! This wastes time and increases error chances. Always check for perfect square patterns first and factor as $(2x + 6)^2 = 0$ .

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

How do I recognize a perfect square trinomial?

Look for the pattern $a^2 + 2ab + b^2$ ! In $4x^2 + 24x + 36$ , we have (2x)² + 2(2x)(6) + 6², which forms the perfect square $(2x + 6)^2$ .

Why does this equation only have one solution?

When a quadratic factors as $(expression)^2 = 0$ , it's called a repeated root. The same solution x = -3 appears twice, so technically there's only one unique answer.

What if I can't see the perfect square pattern?

Start by factoring out common factors first! Here, factor out 4: $4(x^2 + 6x + 9) = 0$ . Then $x^2 + 6x + 9 = (x + 3)^2$ is easier to spot.

Can I solve this using other methods?

Yes! You could use the quadratic formula or completing the square, but factoring perfect squares is the fastest method. All methods will give the same answer: x = -3.

How do I verify my factoring is correct?

Always expand your factored form! Multiply out $(2x + 6)^2$ to get $4x^2 + 24x + 36$ . If it matches the original equation, your factoring is right!

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