Question
7. [-/1 Points] DETAILS MY Factor. \( 4 x^{2}+20 x y+9 y^{2} \) SUBMIT ANSWER 8. [-/1 Points] DETAILS MY
Ask by Hamilton Mcfarlane. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression \(4x^2+20xy+9y^2\) factors to \((2x+9y)(2x+y)\).
Solution
Factor the expression \( 4x^2+20xy+9y^2 \).
Factor the expression by following steps:
- step0: Factor:
\(4x^{2}+20xy+9y^{2}\)
- step1: Rewrite the expression:
\(4x^{2}+\left(2+18\right)xy+9y^{2}\)
- step2: Calculate:
\(4x^{2}+2xy+18yx+9y^{2}\)
- step3: Rewrite the expression:
\(2x\times 2x+2xy+9y\times 2x+9y\times y\)
- step4: Factor the expression:
\(2x\left(2x+y\right)+9y\left(2x+y\right)\)
- step5: Factor the expression:
\(\left(2x+9y\right)\left(2x+y\right)\)
The factored form of the expression \(4x^2+20xy+9y^2\) is \((2x+9y)(2x+y)\).
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Simplify this solution The Deep Dive
Did you know that factoring polynomials dates back to ancient civilizations? The Babylonians were among the first to solve quadratic equations, laying the groundwork for algebra as we know it. The method for handling equations like \(4x^{2} + 20xy + 9y^{2}\) has evolved over the centuries, showcasing the beauty of mathematical development! Factoring in real-world applications can help simplify complex problems. For instance, in physics, equations similar to \(4x^{2} + 20xy + 9y^{2}\) can emerge in projectile motion analysis. By breaking these down into factors, it’s easier to find roots that indicate critical points, such as maximum height or time of flight. It's math in action!
