Question

1. factor the following expression: $6x^2 - 19x + 15$

Explanation:

Step 1: Multiply \(a\) and \(c\)

For the quadratic \(ax^{2}+bx + c\) (here \(a = 6\), \(b=-19\), \(c = 15\)), we calculate \(a\times c=6\times15 = 90\).

Step 2: Find two numbers

We need two numbers that multiply to \(90\) and add up to \(b=-19\). The numbers are \(-9\) and \(-10\) since \((-9)\times(-10)=90\) and \((-9)+(-10)=-19\).

Step 3: Split the middle term

Rewrite the middle term using these two numbers:
\(6x^{2}-9x - 10x + 15\)

Step 4: Group and factor

Group the first two and last two terms:
\((6x^{2}-9x)+(-10x + 15)\)
Factor out the greatest common factor from each group:
\(3x(2x - 3)-5(2x - 3)\)

Step 5: Factor out the common binomial

Factor out \((2x - 3)\):
\((3x - 5)(2x - 3)\)

Answer:

\((3x - 5)(2x - 3)\)