Answer :
Sure! Let's solve the equation [tex]\(\sqrt{4(x-5)^2} = 40\)[/tex] step by step:
1. Eliminate the Square Root:
To remove the square root, square both sides of the equation:
[tex]\[
(\sqrt{4(x-5)^2})^2 = 40^2
\][/tex]
This simplifies to:
[tex]\[
4(x-5)^2 = 1600
\][/tex]
2. Divide by 4:
To simplify further, divide both sides of the equation by 4:
[tex]\[
(x-5)^2 = 400
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, take the square root of both sides to solve for [tex]\(x-5\)[/tex]:
[tex]\[
x-5 = \pm\sqrt{400}
\][/tex]
Since [tex]\(\sqrt{400} = 20\)[/tex], we have:
[tex]\[
x-5 = 20 \quad \text{or} \quad x-5 = -20
\][/tex]
4. Find the Possible Values of [tex]\(x\)[/tex]:
Solve each equation for [tex]\(x\)[/tex]:
- For [tex]\(x - 5 = 20\)[/tex]:
[tex]\[
x = 20 + 5 = 25
\][/tex]
- For [tex]\(x - 5 = -20\)[/tex]:
[tex]\[
x = -20 + 5 = -15
\][/tex]
Thus, the solutions to the equation [tex]\(\sqrt{4(x-5)^2} = 40\)[/tex] are [tex]\(x = 25\)[/tex] and [tex]\(x = -15\)[/tex].
1. Eliminate the Square Root:
To remove the square root, square both sides of the equation:
[tex]\[
(\sqrt{4(x-5)^2})^2 = 40^2
\][/tex]
This simplifies to:
[tex]\[
4(x-5)^2 = 1600
\][/tex]
2. Divide by 4:
To simplify further, divide both sides of the equation by 4:
[tex]\[
(x-5)^2 = 400
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now, take the square root of both sides to solve for [tex]\(x-5\)[/tex]:
[tex]\[
x-5 = \pm\sqrt{400}
\][/tex]
Since [tex]\(\sqrt{400} = 20\)[/tex], we have:
[tex]\[
x-5 = 20 \quad \text{or} \quad x-5 = -20
\][/tex]
4. Find the Possible Values of [tex]\(x\)[/tex]:
Solve each equation for [tex]\(x\)[/tex]:
- For [tex]\(x - 5 = 20\)[/tex]:
[tex]\[
x = 20 + 5 = 25
\][/tex]
- For [tex]\(x - 5 = -20\)[/tex]:
[tex]\[
x = -20 + 5 = -15
\][/tex]
Thus, the solutions to the equation [tex]\(\sqrt{4(x-5)^2} = 40\)[/tex] are [tex]\(x = 25\)[/tex] and [tex]\(x = -15\)[/tex].
