Questions: Determine the factors of the quadratic expression: Factor. 5 x^2+8 x+3

Determine the factors of the quadratic expression: Factor. 5 x^2+8 x+3
Transcript text: Determine the factors of the quadratic expression: Factor. \[ 5 x^{2}+8 x+3 \]
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Solution

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Solution Steps

To factor the quadratic expression \(5x^2 + 8x + 3\), we need to find two numbers that multiply to the product of the coefficient of \(x^2\) (which is 5) and the constant term (which is 3), and add up to the coefficient of \(x\) (which is 8). Once these numbers are found, we can use them to split the middle term and factor by grouping.

Step 1: Identify the Quadratic Expression

We are given the quadratic expression \(5x^2 + 8x + 3\). Our goal is to factor this expression into the product of two binomials.

Step 2: Determine the Product and Sum

To factor the quadratic expression, we need to find two numbers that multiply to the product of the coefficient of \(x^2\) (which is 5) and the constant term (which is 3), i.e., \(5 \times 3 = 15\), and add up to the coefficient of \(x\) (which is 8).

Step 3: Find the Numbers

The numbers that satisfy these conditions are 3 and 5, since \(3 \times 5 = 15\) and \(3 + 5 = 8\).

Step 4: Factor by Grouping

Using these numbers, we can rewrite the middle term \(8x\) as \(3x + 5x\) and factor by grouping: \[ 5x^2 + 8x + 3 = 5x^2 + 3x + 5x + 3 \] Group the terms: \[ = (5x^2 + 3x) + (5x + 3) \] Factor out the common factors in each group: \[ = x(5x + 3) + 1(5x + 3) \] Factor out the common binomial: \[ = (x + 1)(5x + 3) \]

Final Answer

\(\boxed{(x + 1)(5x + 3)}\)

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