Question

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Answer 1

d) \(\left(5x-15\right)\left(x-4\right)=0\)

\(\Leftrightarrow5\left(x-3\right)\left(x-4\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=4\end{matrix}\right.\)

e) \(\left(x+3\right)^2-9^2=12^2\)

\(\Leftrightarrow\left(x+3\right)^2=225\)

\(\Leftrightarrow\left[{}\begin{matrix}x+3=15\\x+3=-15\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=12\\x=-18\end{matrix}\right.\)

f) \(2^x+2^{x+1}=48\)

\(\Leftrightarrow2^x\left(1+2\right)=48\)

\(\Leftrightarrow2^x.3=48\)

\(\Leftrightarrow2^x=16\)

\(\Leftrightarrow x=4\)

Answer 2

d) \(\left(5x-15\right)\left(x-4\right)=0\Leftrightarrow\left[{}\begin{matrix}5x-15=0\\x-4=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=3\\x=4\end{matrix}\right.\)

e)\(\left(x+3\right)^2-9^2=12^2\Leftrightarrow\left(x+3\right)^2=12^2+9^2=225\Leftrightarrow\left[{}\begin{matrix}x+3=15\\x+3=-15\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=12\\x=-18\end{matrix}\right.\)

f)\(2^x+2^{x+1}=48\Leftrightarrow2^x\left(1+2\right)=48\Leftrightarrow2^x.3=48\Leftrightarrow2^x=16\Leftrightarrow2^x=2^4\Leftrightarrow x=4\)

Answer 3

d. (5x - 15)(x - 4) = 0

<=> \(\Leftrightarrow\left[{}\begin{matrix}5x-15=0\\x-4=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=3\\x=4\end{matrix}\right.\)

e. (x + 3)² - 9² = 12²

<=> (x + 3 - 9)(x + 3 + 9) = 12²

<=> (x - 6)(x + 12) = 144

<=> x² + 12x - 6x - 72 = 144

<=> x² + 6x - 72 - 144 = 0

<=> x² + 6x - 216 = 0

<=> x² - 12x + 18x - 216 = 0

<=> x(x - 12) + 18(x - 12) = 0

<=> (x + 18)(x - 12) = 0

<=> \(\left[{}\begin{matrix}x+18=0\\x-12=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-18\\x=12\end{matrix}\right.\)