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Answer :
To solve the equation [tex]\( 5^{2x+2} - 20 \times 5^{2x} = 625 \)[/tex], let's work through it step by step to find the value of [tex]\( x \)[/tex].
1. Rewrite the Equation:
Start by simplifying the equation. We can use a substitution to make it easier to handle. Let [tex]\( y = 5^{2x} \)[/tex]. This substitution gives us:
[tex]\[
5^{2x+2} = (5^2) \cdot 5^{2x} = 25 \cdot 5^{2x} = 25y
\][/tex]
So, the equation becomes:
[tex]\[
25y - 20y = 625
\][/tex]
2. Simplify the Equation:
Simplify the terms:
[tex]\[
5y = 625
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Divide both sides by 5 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{625}{5} = 125
\][/tex]
Since [tex]\( y = 5^{2x} \)[/tex], we have:
[tex]\[
5^{2x} = 125
\][/tex]
4. Express 125 as a Power of 5:
Notice that 125 can be written as:
[tex]\[
125 = 5^3
\][/tex]
Therefore, we have:
[tex]\[
5^{2x} = 5^3
\][/tex]
5. Equalize the Exponents:
Since the bases (5) are the same, we can set the exponents equal to each other:
[tex]\[
2x = 3
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Divide both sides by 2:
[tex]\[
x = \frac{3}{2}
\][/tex]
Therefore, the solution to the equation is [tex]\( x = \frac{3}{2} \)[/tex].
Additionally, there might be complex solutions, but for the purposes of this problem, [tex]\( x = \frac{3}{2} \)[/tex] is the primary real solution.
1. Rewrite the Equation:
Start by simplifying the equation. We can use a substitution to make it easier to handle. Let [tex]\( y = 5^{2x} \)[/tex]. This substitution gives us:
[tex]\[
5^{2x+2} = (5^2) \cdot 5^{2x} = 25 \cdot 5^{2x} = 25y
\][/tex]
So, the equation becomes:
[tex]\[
25y - 20y = 625
\][/tex]
2. Simplify the Equation:
Simplify the terms:
[tex]\[
5y = 625
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Divide both sides by 5 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{625}{5} = 125
\][/tex]
Since [tex]\( y = 5^{2x} \)[/tex], we have:
[tex]\[
5^{2x} = 125
\][/tex]
4. Express 125 as a Power of 5:
Notice that 125 can be written as:
[tex]\[
125 = 5^3
\][/tex]
Therefore, we have:
[tex]\[
5^{2x} = 5^3
\][/tex]
5. Equalize the Exponents:
Since the bases (5) are the same, we can set the exponents equal to each other:
[tex]\[
2x = 3
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Divide both sides by 2:
[tex]\[
x = \frac{3}{2}
\][/tex]
Therefore, the solution to the equation is [tex]\( x = \frac{3}{2} \)[/tex].
Additionally, there might be complex solutions, but for the purposes of this problem, [tex]\( x = \frac{3}{2} \)[/tex] is the primary real solution.
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Rewritten by : Jeany
