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:
\[ \boxed{(2x - 3)(x - 5)} \]
1 answer