Factor completely: 2x² + 9x + 10
Understand the Problem
The question is asking us to factor the quadratic expression 2x² + 9x + 10 completely. This involves identifying two binomials that, when multiplied together, yield the original expression.
Answer
$$(2x + 5)(x + 2)$$
Answer for screen readers
The completely factored form of the expression $2x^2 + 9x + 10$ is: $$(2x + 5)(x + 2)$$
Steps to Solve
- Identify the coefficients of the quadratic equation
The standard form of a quadratic expression is $ax^2 + bx + c$. For $2x^2 + 9x + 10$, we have:
- $a = 2$
- $b = 9$
- $c = 10$
- Find two numbers that multiply to $a \cdot c$ and add to $b$
First, calculate $a \cdot c$: $$ a \cdot c = 2 \cdot 10 = 20 $$
Next, we need to find two numbers that multiply to $20$ and add up to $9$. The numbers that meet these criteria are $4$ and $5$:
- $4 \cdot 5 = 20$
- $4 + 5 = 9$
- Rewrite the middle term
Using the numbers found above, rewrite the expression: $$ 2x^2 + 4x + 5x + 10 $$
- Group the terms
Group the first two and the last two terms: $$ (2x^2 + 4x) + (5x + 10) $$
- Factor out the greatest common factor from each group
From the first group, factor out $2x$, and from the second group, factor out $5$: $$ 2x(x + 2) + 5(x + 2) $$
- Factor out the common binomial
Now, factor out the common binomial $(x + 2)$: $$ (2x + 5)(x + 2) $$
The completely factored form of the expression $2x^2 + 9x + 10$ is: $$(2x + 5)(x + 2)$$
More Information
The process of factoring involves breaking down polynomials into simpler binomial factors. This can be useful in simplifying equations or solving them without a calculator.
Tips
- Forgetting to factor out the greatest common factor at the beginning.
- Misidentifying the two numbers that multiply and add correctly.
- Not rewriting the middle term correctly, which can lead to errors in grouping.
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