Certainly! Let's evaluate the expression [tex]\( 3x^2 + 7xy - 6y^2 \)[/tex] step by step, given that [tex]\( x = 2 \)[/tex] and [tex]\( y = -3 \)[/tex].
1. Substitute the given values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] into the expression:
[tex]\[
3(2)^2 + 7(2)(-3) - 6(-3)^2
\][/tex]
2. Evaluate each term separately:
- First term: [tex]\( 3(2)^2 \)[/tex]
[tex]\[
(2)^2 = 4, \quad \text{then} \quad 3 \times 4 = 12
\][/tex]
- Second term: [tex]\( 7(2)(-3) \)[/tex]
[tex]\[
2 \times (-3) = -6, \quad \text{then} \quad 7 \times -6 = -42
\][/tex]
- Third term: [tex]\( -6(-3)^2 \)[/tex]
[tex]\[
(-3)^2 = 9, \quad \text{then} \quad -6 \times 9 = -54
\][/tex]
3. Sum the results of the evaluated terms:
[tex]\[
12 + (-42) + (-54)
\][/tex]
4. Performing the addition step-by-step:
- Start with the first two terms: [tex]\( 12 + (-42) \)[/tex]
[tex]\[
12 - 42 = -30
\][/tex]
- Add the third term: [tex]\( -30 + (-54) \)[/tex]
[tex]\[
-30 - 54 = -84
\][/tex]
Therefore, the result of evaluating the expression [tex]\( 3x^2 + 7xy - 6y^2 \)[/tex] with [tex]\( x = 2 \)[/tex] and [tex]\( y = -3 \)[/tex] is [tex]\(\boxed{-84}\)[/tex].
