Factor completely.
2, x, squared, plus, 5, x, minus, 3
2x
2
+5x−3

Let's factor the quadratic expression \(2x^2 + 5x - 3\).

### Step 1: Multiply the coefficient of \(x^2\) and the constant term.
Here, the coefficient of \(x^2\) is 2, and the constant term is \(-3\). Multiply them together:
\[
2 \times (-3) = -6
\]

### Step 2: Find two numbers that multiply to \(-6\) and add to 5.
We need two numbers whose product is \(-6\) and whose sum is 5. These numbers are 6 and \(-1\) because:
\[
6 \times (-1) = -6 \quad \text{and} \quad 6 + (-1) = 5
\]

### Step 3: Rewrite the middle term using 6 and \(-1\).
We can rewrite the quadratic as:
\[
2x^2 + 6x - x - 3
\]

### Step 4: Factor by grouping.
Group the terms in pairs:
\[
(2x^2 + 6x) - (x + 3)
\]
Factor each group:
\[
2x(x + 3) - 1(x + 3)
\]

### Step 5: Factor out the common binomial.
\[
(2x - 1)(x + 3)
\]

So, the fully factored form of the quadratic expression \(2x^2 + 5x - 3\) is:
\[
(2x - 1)(x + 3)
\]

Let me know if you want more details or have any questions!

Here are 5 related questions to deepen your understanding:
1. How do you factor trinomials when the coefficient of \(x^2\) is not 1?
2. Can all quadratics be factored, or are there some that can only be solved using the quadratic formula?
3. What are some shortcuts for recognizing common factorable patterns in quadratic expressions?
4. How would you check if the factors of a quadratic expression are correct?
5. How does completing the square compare to factoring for solving quadratic equations?

**Tip:** Always double-check your factorization by expanding the factors to ensure they give the original expression!

Learn how to factor the quadratic expression 2x^2 + 5x - 3 with a clear, step-by-step solution. This guide covers factoring by grouping and key algebraic techniques, perfect for students in Grades 8-10 looking to strengthen their understanding of quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation