Solve the Quadratic Equation: x² + 10x + 16 = 0

Q: How do I find the two numbers that work for factoring?

Look for two numbers that multiply to give the constant term (16) and add to give the middle coefficient (10). Try factor pairs of 16: 1×16, 2×8, 4×4. Since 2+8=10, use 2 and 8!

Q: Why are the solutions negative when the factors have positive numbers?

When you have $ (x+2)(x+8) = 0 $, you solve x+2=0 and x+8=0. This gives x = -2 and x = -8. The plus signs in factors create negative solutions!

Q: How do I check if my factoring is correct?

Expand your factors back out: $ (x+2)(x+8) = x^2 + 8x + 2x + 16 = x^2 + 10x + 16 $. If it matches the original equation, your factoring is right!

Q: Do both solutions always work in the original equation?

Yes, always check both! Substitute each solution back into the original equation. If both make the equation true (equal zero), then both are correct solutions.

Quadratic Equations with Quick Factoring

$x^2+10x+16=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's find X together.

00:08 First, break it down using a trinomial. Focus on the coefficients.

00:13 We need two numbers that add up to B, which is 10.

00:18 And their product should be C, which is 16.

00:24 These are our matching numbers. Let's substitute them in parentheses.

00:31 Next, we'll find what makes each factor equal to zero.

00:41 And that's how we solve this problem!

Step-by-step written solution

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Understand the problem

$x^2+10x+16=0$

Step-by-step solution

Let's observe that the given equation:

$x^2+10x+16=0$ is a quadratic equation that can be solved using quick factoring:

$x^2+10x+16=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=16\\ \underline{?}+\underline{?}=10\end{cases}\\ \downarrow\\ (x+2)(x+8)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x+2)(x+8)=0 \\ \downarrow\\ x+2=0\rightarrow\boxed{x=-2}\\ x+8=0\rightarrow\boxed{x=-8}\\ \boxed{x=-2,-8}$ Therefore, the correct answer is answer B.

Final Answer

$x=-8,x=-2$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Find two numbers that multiply to 16 and add to 10
Technique: $2 \times 8 = 16$ and $2 + 8 = 10$ , so factors are (x+2)(x+8)
Check: Substitute $x = -2$ : $(-2)^2 + 10(-2) + 16 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong signs in factors
Don't write (x-2)(x-8) when you need positive factors = wrong signs in solutions! The equation x² + 10x + 16 has positive coefficients, so both factors need positive constants. Always check: your factors should give +2 and +8, making solutions x = -2 and x = -8.

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 find the two numbers that work for factoring?

Look for two numbers that multiply to give the constant term (16) and add to give the middle coefficient (10). Try factor pairs of 16: 1×16, 2×8, 4×4. Since 2+8=10, use 2 and 8!

Why are the solutions negative when the factors have positive numbers?

When you have $(x+2)(x+8) = 0$ , you solve x+2=0 and x+8=0. This gives x = -2 and x = -8. The plus signs in factors create negative solutions!

What if I can't factor the quadratic easily?

Not all quadratics factor nicely! If you can't find two integers that work, you can use the quadratic formula or completing the square method instead.

How do I check if my factoring is correct?

Expand your factors back out: $(x+2)(x+8) = x^2 + 8x + 2x + 16 = x^2 + 10x + 16$ . If it matches the original equation, your factoring is right!

Do both solutions always work in the original equation?

Yes, always check both! Substitute each solution back into the original equation. If both make the equation true (equal zero), then both are correct solutions.

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