Let's go through the factoring process step by step for the expression 2x^2 - 16x + 30:
Step 1: Divide everything by 2 to simplify the expression:
(2x^2)/2 - (16x)/2 + 30/2
Simplify:
x^2 - 8x + 15
Step 2: Factor the quadratic expression x^2 - 8x + 15.
To factor a quadratic expression of the form ax^2 + bx + c, where a, b, and c are constants, we look for two binomials in the form (px + q)(rx + s) that multiply together to give the original expression. In this case, a = 1 (since the coefficient of x^2 is 1), b = -8, and c = 15.
Now, we need to find two numbers that:
Multiply to give the product of a and c: (1 * 15 = 15)
Add to give the coefficient of x, which is b (in this case, -8)
The two numbers that satisfy these conditions are -5 and -3, as:
-5 * -3 = 15 (which is the product of 1 and 15), and
-5 + (-3) = -8 (which is the coefficient of x, b).
Step 3: Rewrite the middle term (-8x) using the two numbers found in Step 2.
x^2 - 8x + 15 can be rewritten as:
x^2 - 5x - 3x + 15
Step 4: Group the terms and factor by grouping.
Factor out a common term from the first two terms and the last two terms:
x(x - 5) - 3(x - 5)
Step 5: Factor out the common binomial (x - 5).
(x - 5)(x - 3)
Step 6: Lastly, bring back the common factor of 2 that we divided earlier:
2(x - 5)(x - 3)
So, the correct factored form of the quadratic expression 2x^2 - 16x + 30 is 2(x - 5)(x - 3).