Questions: Factor. 14x^2+27x-20

Transcript text: Factor. \[ 14 x^{2}+27 x-20 \]

Solution

Solution Steps

To factor the quadratic expression \(14x^2 + 27x - 20\), we can use the method of finding two numbers that multiply to the product of the leading coefficient and the constant term (i.e., \(14 \times -20\)) and add to the middle coefficient (i.e., 27). Then, we can use these numbers to split the middle term and factor by grouping.

Step 1: Identify the Quadratic Expression

We are given the quadratic expression: \[ 14x^2 + 27x - 20 \]

Step 2: Factor the Expression

To factor the expression, we look for two binomials \((ax + b)(cx + d)\) such that:

\(ac = 14\)
\(bd = -20\)
\(ad + bc = 27\)

Step 3: Verify the Factorization

The factorization of the expression is: \[ (2x + 5)(7x - 4) \]

Step 4: Expand to Confirm

Expanding \((2x + 5)(7x - 4)\) gives: \[ 2x \cdot 7x + 2x \cdot (-4) + 5 \cdot 7x + 5 \cdot (-4) \] \[ = 14x^2 - 8x + 35x - 20 \] \[ = 14x^2 + 27x - 20 \]

This confirms the factorization is correct.

Final Answer

The factorization of the quadratic expression is: \[ \boxed{(2x + 5)(7x - 4)} \]

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