Solve the Quadratic Equation: x^2 + 10x - 24 = 0

Q: How do I know which two numbers multiply to -24 and add to 10?

List all factor pairs of -24: (1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), (-4, 6). Then check which pair adds to positive 10. Only 12 + (-2) = 10 works!

Q: Why does (x + 12)(x - 2) = 0 give me x = -12 and x = 2?

When a product equals zero, at least one factor must be zero. So either x + 12 = 0 (giving x = -12) or x - 2 = 0 (giving x = 2). This is called the Zero Product Property.

Q: What if I can't find factor pairs that work?

If factoring seems impossible, try the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. But for most textbook problems, factoring will work if you're patient with finding the right factor pairs.

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

Use FOIL to expand your factors back to the original equation. $ (x + 12)(x - 2) = x^2 - 2x + 12x - 24 = x^2 + 10x - 24 $ ✓

Q: Do I always get two different solutions for quadratic equations?

Usually yes, but not always! Sometimes you get one repeated solution (like x = 3, x = 3) or even no real solutions if the discriminant is negative. This problem has two distinct real solutions.

Quadratic Factoring with Integer Solutions

$x^2+10x-24=0$

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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, examining coefficients

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

00:13 and their product equals C (-24)

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

00:30 Find what zeroes 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

$x^2+10x-24=0$

Step-by-step solution

Let's observe that the given equation:

$x^2+10x-24=0$ is a quadratic equation that can be solved using quick factoring:

$x^2+10x-24=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=-24\\ \underline{?}+\underline{?}=10\end{cases}\\ \downarrow\\ (x+12)(x-2)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x+12)(x-2)=0 \\ \downarrow\\ x+12=0\rightarrow\boxed{x=-12}\\ x-2=0\rightarrow\boxed{x=2}\\ \boxed{x=-12,2}$ Therefore, the correct answer is answer B.

Final Answer

$x=2,x=-12$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Find two numbers that multiply to $c$ and add to $b$
Factor Pair: For $-24$ and $+10$ , use $12 \times (-2) = -24$ and $12 + (-2) = 10$
Verification: Substitute $x = 2$ : $(2)^2 + 10(2) - 24 = 4 + 20 - 24 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Getting the signs wrong when factoring
Don't write (x - 12)(x + 2) when you need (x + 12)(x - 2) = wrong signs on solutions! Students often mix up which factor gets the positive or negative sign. Always check: the factors should multiply to give the original constant term and add to give the 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 multiply to -24 and add to 10?

List all factor pairs of -24: (1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), (-4, 6). Then check which pair adds to positive 10. Only 12 + (-2) = 10 works!

Why does (x + 12)(x - 2) = 0 give me x = -12 and x = 2?

When a product equals zero, at least one factor must be zero. So either x + 12 = 0 (giving x = -12) or x - 2 = 0 (giving x = 2). This is called the Zero Product Property.

What if I can't find factor pairs that work?

If factoring seems impossible, try the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . But for most textbook problems, factoring will work if you're patient with finding the right factor pairs.

How can I check my factored form is correct?

Use FOIL to expand your factors back to the original equation. $(x + 12)(x - 2) = x^2 - 2x + 12x - 24 = x^2 + 10x - 24$ ✓

Do I always get two different solutions for quadratic equations?

Usually yes, but not always! Sometimes you get one repeated solution (like x = 3, x = 3) or even no real solutions if the discriminant is negative. This problem has two distinct real solutions.

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