Question
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Tutor-Verified Answer
Answer
\( 3p^2 + 7pq - 6q^2 = (p + 3q)(3p - 2q) \)
Solution
1. Start with the expression:
\[
3p^2 + 7pq - 6q^2
\]
2. To factor, consider splitting the middle term. Look for two numbers that multiply to \( 3 \times (-6q^2) = -18q^2 \) and add to \( 7q \). The numbers \( 9q \) and \( -2q \) satisfy this since:
\[
9q + (-2q) = 7q \quad \text{and} \quad 9q \times (-2q) = -18q^2.
\]
3. Rewrite the middle term using these numbers:
\[
3p^2 + 9pq - 2pq - 6q^2.
\]
4. Group the terms:
\[
(3p^2 + 9pq) + (-2pq - 6q^2).
\]
5. Factor out the common factors from each group:
\[
3p(p + 3q) - 2q(p + 3q).
\]
6. Factor out the common binomial \((p + 3q)\):
\[
(p + 3q)(3p - 2q).
\]
Thus, the factored form of \( 3p^2 + 7pq - 6q^2 \) is:
\[
(p + 3q)(3p - 2q)
\]
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Simplify this solution Bonus Knowledge
Did you know that the expression \( 3 p^{2}+7 p q-6 q^{2} \) can be factored using the method of grouping? To factor it, you look for two numbers that multiply to \( (3)(-6) = -18 \) and add to \( 7 \). Those numbers are \( 9 \) and \( -2 \). This can lead you to rearranging and grouping the terms effectively! For a real-world application, such quadratic expressions are frequently encountered in areas such as physics—think projectile motion! This type of expression can represent factors like time and distance, making them essential in calculating trajectories. By mastering these concepts, you can solve practical problems that involve kinetic and potential energy!
