Simplify 8m - (2x5m) + 6m + (x+3y)

Mathematics
Simplify 8m - (2x5m) + 6m + (x+3y)

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Answer

34

Step 1: Expand (5cos⁡x−3sin⁡x)2(5\cos x - 3\sin x)^2.

(5cos⁡x−3sin⁡x)2=25cos⁡2x−2⋅5cos⁡x⋅3sin⁡x+9sin⁡2x=25cos⁡2x−30cos⁡xsin⁡x+9sin⁡2x\begin{align*} (5\cos x - 3\sin x)^2 &= 25\cos^2 x - 2\cdot5\cos x\cdot3\sin x + 9\sin^2 x \\ &= 25\cos^2 x - 30\cos x\sin x + 9\sin^2 x \end{align*}

Step 2: Expand (3cos⁡x+5sin⁡x)2(3\cos x + 5\sin x)^2.

(3cos⁡x+5sin⁡x)2=9cos⁡2x+2⋅3cos⁡x⋅5sin⁡x+25sin⁡2x=9cos⁡2x+30cos⁡xsin⁡x+25sin⁡2x\begin{align*} (3\cos x + 5\sin x)^2 &= 9\cos^2 x + 2\cdot3\cos x\cdot5\sin x + 25\sin^2 x \\ &= 9\cos^2 x + 30\cos x\sin x + 25\sin^2 x \end{align*}

Step 3: Add the expansions.

(5cos⁡x−3sin⁡x)2+(3cos⁡x+5sin⁡x)2=(25cos⁡2x−30cos⁡xsin⁡x+9sin⁡2x)+(9cos⁡2x+30cos⁡xsin⁡x+25sin⁡2x)\begin{align*} &(5\cos x - 3\sin x)^2 + (3\cos x + 5\sin x)^2 \\ &= (25\cos^2 x - 30\cos x\sin x + 9\sin^2 x) + (9\cos^2 x + 30\cos x\sin x + 25\sin^2 x) \end{align*}

Step 4: Combine like terms.

=25cos⁡2x+9cos⁡2x+9sin⁡2x+25sin⁡2x−30cos⁡xsin⁡x+30cos⁡xsin⁡x=34cos⁡2x+34sin⁡2x+0=34(cos⁡2x+sin⁡2x)\begin{align*} &= 25\cos^2 x + 9\cos^2 x + 9\sin^2 x + 25\sin^2 x - 30\cos x\sin x + 30\cos x\sin x \\ &= 34\cos^2 x + 34\sin^2 x + 0 \\ &= 34(\cos^2 x + \sin^2 x) \end{align*}

Step 5: Use the Pythagorean identity.

cos⁡2x+sin⁡2x=1\cos^2 x + \sin^2 x = 1 34(cos⁡2x+sin⁡2x)=34⋅1=3434(\cos^2 x + \sin^2 x) = 34 \cdot 1 = 34

Final answer: 34

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Quick Answer

Expand (5 x - 3 x)^2. (5 x - 3 x)^2 &= 25^2 x - 2·5 x·3 x + 9^2 x \\ &= 25^2 x - 30 x x + 9^2 x Step 2: Expand (3 x + 5 x)^2.

Simplify 8m - (2x5m) + 6m + (x+3y)
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Expand (5 x - 3 x)^2. (5 x - 3 x)^2 &= 25^2 x - 2·5 x·3 x + 9^2 x \\ &= 25^2 x - 30 x x + 9^2 x Step 2: Expand (3 x + 5 x)^2. (3 x + 5 x)^2 &= 9^2 x + 2·3 x·5 x + 25^2 x \\ &= 9^2 x + 30 x x + 25^2 x Step 3: Add the expansions. &(5 x - 3 x)^2 + (3 x + 5 x)^2 \\ &= (25^2 x - 30 x x + 9^2 x) + (9^2 x + 30 x x + 25^2 x) Step 4: Combine like terms. &= 25^2 x + 9^2 x + 9^2 x + 25^2 x - 30 x x + 30 x x \\ &= 34^2 x + 34^2 x + 0 \\ &= 34(^2 x + ^2 x) Step 5: Use the Pythagorean identity. ^2 x + ^2 x = 1 34(^2 x + ^2 x) = 34 · 1 = 34 Final answer: 34