Question

Factor completely.
3, x, squared, minus, 8, x, plus, 5
3x
2
−8x+5

Answer 1

To factor the quadratic expression \(3x^2 - 8x + 5\) completely, we can use the method of finding two numbers that multiply to the product of the leading coefficient (3) and the constant term (5), which is \(3 \times 5 = 15\), and also add up to the middle coefficient (-8).

The two numbers that satisfy these conditions are -3 and -5 since:
- Multiplying: \(-3 \times -5 = 15\)
- Adding: \(-3 + -5 = -8\)

Now we can rewrite the middle term \(-8x\) using -3 and -5:
\[
3x^2 - 3x - 5x + 5
\]

Next, we group the terms:
\[
(3x^2 - 3x) + (-5x + 5)
\]

Now we factor out the common factors in each group:
\[
3x(x - 1) - 5(x - 1)
\]

Since \((x - 1)\) is common in both groups, we can factor it out:
\[
(3x - 5)(x - 1)
\]

Thus, the complete factorization of the expression \(3x^2 - 8x + 5\) is:
\[
\boxed{(3x - 5)(x - 1)}
\]