Asked by pp
Factor completely.
2, x, squared, minus, 13, x, plus, 15
2x
2
−13x+15
2, x, squared, minus, 13, x, plus, 15
2x
2
−13x+15
Answers
Answered by
GPT-4o mini
To factor the quadratic expression \( 2x^2 - 13x + 15 \), we can use the method of factoring by grouping or the quadratic formula. Let's find two numbers that multiply to \( 2 \times 15 = 30 \) and add to \(-13\).
The two numbers that fit this are \(-3\) and \(-10\) since:
\[
-3 \times -10 = 30 \quad \text{and} \quad -3 + (-10) = -13
\]
Now, we can rewrite the middle term \(-13x\) as \(-3x - 10x\):
\[
2x^2 - 3x - 10x + 15
\]
Next, we can group the terms:
\[
(2x^2 - 3x) + (-10x + 15)
\]
Factoring out the common factors from each group:
\[
x(2x - 3) - 5(2x - 3)
\]
Now, we can factor out the common binomial factor \((2x - 3)\):
\[
(2x - 3)(x - 5)
\]
Thus, the completely factored form of the expression \( 2x^2 - 13x + 15 \) is:
\[
\boxed{(2x - 3)(x - 5)}
\]
The two numbers that fit this are \(-3\) and \(-10\) since:
\[
-3 \times -10 = 30 \quad \text{and} \quad -3 + (-10) = -13
\]
Now, we can rewrite the middle term \(-13x\) as \(-3x - 10x\):
\[
2x^2 - 3x - 10x + 15
\]
Next, we can group the terms:
\[
(2x^2 - 3x) + (-10x + 15)
\]
Factoring out the common factors from each group:
\[
x(2x - 3) - 5(2x - 3)
\]
Now, we can factor out the common binomial factor \((2x - 3)\):
\[
(2x - 3)(x - 5)
\]
Thus, the completely factored form of the expression \( 2x^2 - 13x + 15 \) is:
\[
\boxed{(2x - 3)(x - 5)}
\]