Solve the Quadratic Equation: x² + 9x + 20 = 0

Q: How do I know which two numbers to use for factoring?

Look for two numbers that do two jobs: multiply to give the constant term (20) and add to give the middle coefficient (9). For this problem, 5 and 4 work because 5×4=20 and 5+4=9!

Q: Why are both solutions negative?

When we factor to $ (x+5)(x+4) = 0 $, we solve x+5=0 and x+4=0. This gives us x=-5 and x=-4. The plus signs in the factors create negative solutions!

Q: Can I use the quadratic formula instead?

Absolutely! The quadratic formula $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $ always works. But factoring is faster when the quadratic factors nicely like this one does.

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

Expand your factors using FOIL! $ (x+5)(x+4) = x^2 + 4x + 5x + 20 = x^2 + 9x + 20 $. If you get back the original equation, your factoring is perfect!

Quadratic Equations with Factoring Method

$x^2+9x+20=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Let's break it down using trinomial, let's look at the coefficients

00:09 We want to find 2 numbers whose sum equals B (9)

00:15 and their product equals C (20)

00:24 These are the matching numbers, let's substitute in parentheses

00:31 Let's find what zeros each factor

00:42 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

$x^2+9x+20=0$

Step-by-step solution

Let's observe that the given equation:

$x^2+9x+20=0$ is a quadratic equation that can be solved using quick factoring:

$x^2+9x+20=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=20\\ \underline{?}+\underline{?}=9\end{cases}\\ \downarrow\\ (x+5)(x+4)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x+5)(x+4)=0 \\ \downarrow\\ x+5=0\rightarrow\boxed{x=-5}\\ x+4=0\rightarrow\boxed{x=-4}\\ \boxed{x=-4,-5}$ Therefore, the correct answer is answer A.

Final Answer

$x=-4,x=-5$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Find two numbers that multiply to 20 and add to 9
Technique: Check factors: 5 × 4 = 20, 5 + 4 = 9, so $(x+5)(x+4) = 0$
Check: Substitute $x = -4: (-4)^2 + 9(-4) + 20 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Finding factors that multiply to 20 but ignoring the sum requirement
Don't just find any factors of 20 like 1×20 or 2×10 = wrong factorization! This ignores that the middle term coefficient (9) must equal the sum of your factors. Always check both: factors multiply to constant term AND add to middle coefficient.

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 which two numbers to use for factoring?

Look for two numbers that do two jobs: multiply to give the constant term (20) and add to give the middle coefficient (9). For this problem, 5 and 4 work because 5×4=20 and 5+4=9!

Why are both solutions negative?

When we factor to $(x+5)(x+4) = 0$ , we solve x+5=0 and x+4=0. This gives us x=-5 and x=-4. The plus signs in the factors create negative solutions!

Can I use the quadratic formula instead?

Absolutely! The quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ always works. But factoring is faster when the quadratic factors nicely like this one does.

How do I check if my factoring is correct?

Expand your factors using FOIL! $(x+5)(x+4) = x^2 + 4x + 5x + 20 = x^2 + 9x + 20$ . If you get back the original equation, your factoring is perfect!

What if I can't find two numbers that work?

Not all quadratics factor nicely with integers. If you can't find two whole numbers that multiply to the constant and add to the middle coefficient, use the quadratic formula or completing the square instead.

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