Solve the Quadratic Equation: x²+6x+9=0 Step by Step

Q: How do I know if a trinomial is a perfect square?

Check if the middle term equals twice the product of the square roots of the first and last terms. For $ x^2 + 6x + 9 $: first term = x², last term = 3², middle should be 2(x)(3) = 6x ✓

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

You can always use the quadratic formula or try regular factoring methods! Perfect square recognition is just a shortcut, but other methods will give the same answer.

Q: Can I solve this without factoring?

Absolutely! You could use the quadratic formula with a=1, b=6, c=9. You'll get the same answer: x = -3, but factoring is faster when you recognize the pattern.

Q: How do I check my factored form is correct?

Expand it back! $ (x+3)^2 = (x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9 $ ✓ If it matches the original, your factoring is right!

Perfect Square Trinomials with Factoring Methods

$x^2+6x+9=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Break down 9 to 3 squared

00:09 Factorize 6 into 2 and 3

00:17 Use trinomial and find the product

00:22 Find the zeros of the brackets

00:27 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

$x^2+6x+9=0$

Step-by-step solution

Let's solve the given equation:

$x^2+6x+9=0$ We can identify that the expression on the left side can be factored using the perfect square trinomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$ Let's do this:

$x^2+6x+9=0 \\ x^2\textcolor{blue}{+6x}+3^2=0 \\ x^2\textcolor{blue}{+2\cdot x\cdot3}+3^2=0 \\ \downarrow\\ (x+3)^2=0$ We emphasize that factoring using the mentioned formula was possible only because the middle term in the expression (which is in first power in this case and highlighted in blue in the previous calculation) indeed matched the middle term in the perfect square trinomial formula,

We'll continue and solve the resulting equation, which we'll do using square root extraction on both sides:

$(x+3)^2=0 \hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\\ x+3=0\\ \boxed{x=-3}$ Therefore, the correct answer is answer B.

Final Answer

$x=-3$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Identify $a^2 + 2ab + b^2 = (a+b)^2$ form
Factoring: $x^2 + 6x + 9 = (x+3)^2$ using perfect square formula
Verification: Substitute x = -3: $(-3)^2 + 6(-3) + 9 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly identifying the middle term in perfect square trinomials
Don't assume every trinomial is a perfect square without checking the middle term = wrong factorization! The middle term must equal 2ab where a and b are from the first and last terms. Always verify that 6x = 2(x)(3) before factoring as (x+3)².

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

How do I know if a trinomial is a perfect square?

Check if the middle term equals twice the product of the square roots of the first and last terms. For $x^2 + 6x + 9$ : first term = x², last term = 3², middle should be 2(x)(3) = 6x ✓

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

You can always use the quadratic formula or try regular factoring methods! Perfect square recognition is just a shortcut, but other methods will give the same answer.

Why does (x+3)² = 0 give only one solution?

When a perfect square equals zero, both factors are the same: $(x+3)(x+3) = 0$ . This means x + 3 = 0 twice, so we get one repeated solution: x = -3.

Can I solve this without factoring?

Absolutely! You could use the quadratic formula with a=1, b=6, c=9. You'll get the same answer: x = -3, but factoring is faster when you recognize the pattern.

How do I check my factored form is correct?

Expand it back! $(x+3)^2 = (x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9$ ✓ If it matches the original, your factoring is right!

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