Questions: Factor. 15x^2+13x+2

Transcript text: Factoring a quadratic wi Factor. \[ 15 x^{2}+13 x+2 \]

Solution

Solution Steps

To factor the quadratic expression \(15x^2 + 13x + 2\), we need to find two numbers that multiply to the product of the leading coefficient (15) and the constant term (2), which is 30, and add up to the middle coefficient (13). Once we find these numbers, we can use them to split the middle term and factor by grouping.

Step 1: Identify the Quadratic Expression

We start with the quadratic expression: \[ 15x^2 + 13x + 2 \]

Step 2: Factor the Expression

To factor the expression, we find two numbers that multiply to \(15 \times 2 = 30\) and add up to \(13\). The numbers \(3\) and \(10\) satisfy these conditions. We can rewrite the middle term \(13x\) as \(3x + 10x\): \[ 15x^2 + 3x + 10x + 2 \]

Step 3: Group and Factor

Next, we group the terms: \[ (15x^2 + 3x) + (10x + 2) \] Factoring out the common factors in each group gives us: \[ 3x(5x + 1) + 2(5x + 1) \] Now, we can factor out the common binomial \((5x + 1)\): \[ (3x + 2)(5x + 1) \]

Final Answer

Thus, the factored form of the quadratic expression \(15x^2 + 13x + 2\) is: \[ \boxed{(3x + 2)(5x + 1)} \]

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