Solve x²+10x+9=0: Finding the Number of Solutions

Q: How do I know this equation has exactly two solutions?

Since this is a quadratic equation (highest power is x²), it can have at most 2 solutions. When you factor successfully into two different linear factors like $ (x+1)(x+9) $, you get exactly 2 solutions.

Q: What if I can't factor the quadratic easily?

If factoring seems difficult, you can always use the quadratic formula or complete the square. But for this problem, look for two numbers that multiply to 9 and add to 10: that's 1 and 9!

Q: Why do both factors need to equal zero?

This uses the zero product property: if two things multiply to give zero, then at least one of them must be zero. So if $ (x+1)(x+9) = 0 $, then either x+1=0 or x+9=0 (or both).

Q: Can a quadratic equation have more than two solutions?

No! A quadratic equation can have at most 2 solutions. It might have 2 different solutions (like this problem), 1 repeated solution, or no real solutions at all.

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

Expand your factored form back out! $ (x+1)(x+9) = x² + 9x + x + 9 = x² + 10x + 9 $. If it matches the original equation, your factoring is correct.

Quadratic Equations with Factoring Method

How many solutions does the equation have?

$x^2+10x+9=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

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

00:07 We want to find 2 numbers whose sum equals B (10)

00:14 and their product equals C (9)

00:21 These are the matching numbers, let's substitute in parentheses

00:27 Let's find what zeroes each factor

00:31 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

How many solutions does the equation have?

$x^2+10x+9=0$

Step-by-step solution

Let's observe that the given equation:

$x^2+10x+9=0$ is a quadratic equation that can be solved using quick factoring:

$x^2+10x+9=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=9\\ \underline{?}+\underline{?}=10\end{cases}\\ \downarrow\\ (x+1)(x+9)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x+1)(x+9)=0\\ \downarrow\\ x+1=0\rightarrow\boxed{x=-1}\\ x+9=0\rightarrow\boxed{x=-9}\\ \boxed{x=-1,-9}$ and therefore the given equation has two solutions,

Thus, the correct answer is answer B.

Final Answer

Two solutions

Key Points to Remember

Essential concepts to master this topic

Rule: Quadratic equations always have zero, one, or two solutions
Technique: Factor using two numbers that multiply to 9 and add to 10
Check: Substitute x = -1 and x = -9 back: (-1)² + 10(-1) + 9 = 0 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the zero product property after factoring
Don't just factor to (x+1)(x+9) and stop there = incomplete solution! Factoring only reorganizes the equation but doesn't solve it. Always set each factor equal to zero: x+1=0 and x+9=0 to find both solutions.

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 this equation has exactly two solutions?

Since this is a quadratic equation (highest power is x²), it can have at most 2 solutions. When you factor successfully into two different linear factors like $(x+1)(x+9)$ , you get exactly 2 solutions.

What if I can't factor the quadratic easily?

If factoring seems difficult, you can always use the quadratic formula or complete the square. But for this problem, look for two numbers that multiply to 9 and add to 10: that's 1 and 9!

Why do both factors need to equal zero?

This uses the zero product property: if two things multiply to give zero, then at least one of them must be zero. So if $(x+1)(x+9) = 0$ , then either x+1=0 or x+9=0 (or both).

Can a quadratic equation have more than two solutions?

No! A quadratic equation can have at most 2 solutions. It might have 2 different solutions (like this problem), 1 repeated solution, or no real solutions at all.

How do I check if my factoring is correct?

Expand your factored form back out! $(x+1)(x+9) = x² + 9x + x + 9 = x² + 10x + 9$ . If it matches the original equation, your factoring is correct.

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