Factor k^2 - 2k - 24.
Understand the Problem
The question is asking to factor the quadratic expression k^2 - 2k - 24. To solve this, we will look for two numbers that multiply to -24 and add up to -2, which will help us rewrite the expression in its factored form.
Answer
The factored form of $k^2 - 2k - 24$ is $(k - 6)(k + 4)$.
Answer for screen readers
The factored form of the quadratic expression $k^2 - 2k - 24$ is $(k - 6)(k + 4)$.
Steps to Solve
- Identify the coefficients The quadratic expression is in the form $ax^2 + bx + c$. Here, we have:
- $a = 1$ (coefficient of $k^2$)
- $b = -2$ (coefficient of $k$)
- $c = -24$ (constant term)
- Find two numbers that multiply and add to required values We need to find two numbers that multiply to $c = -24$ and add to $b = -2$.
The numbers that satisfy this are $-6$ and $4$, since:
$$ -6 \times 4 = -24 $$ $$ -6 + 4 = -2 $$
- Rewrite the expression using the found numbers We can rewrite the original expression by breaking down the middle term (which is $-2k$) using the numbers $-6$ and $4$:
$$ k^2 - 6k + 4k - 24 $$
- Factor by grouping Now, we group the terms:
$$ (k^2 - 6k) + (4k - 24) $$
Factor out the common factors from each group:
$$ k(k - 6) + 4(k - 6) $$
- Factor out the common binomial Now, we can factor out the common binomial $(k - 6)$:
$$ (k - 6)(k + 4) $$
The factored form of the quadratic expression $k^2 - 2k - 24$ is $(k - 6)(k + 4)$.
More Information
Factoring quadratics is a fundamental skill in algebra that helps in solving equations and understanding the behavior of parabolas. The numbers found during the factoring process must meet the specific conditions of adding and multiplying to the coefficients of the quadratic expression.
Tips
- Not checking that the two numbers found multiply to the correct value of $c$. It's essential to verify both conditions (addition and multiplication) are satisfied.
- Forgetting to write the correct factored format. Always include parentheses properly.
