Question

5*​15^​x-​3*​5^​(x+​1)-​3^​x+​3=0

Answer 1

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Answer 2

To solve the equation $5 \cdot 15^{x} - 3 \cdot 5^{x+1} - 3^{x} + 3 = 0$, we need to find the value of $x$ that satisfies the equation.

Step 1: Simplify the terms. First, let's rewrite the equation using exponents to simplify it:

$5 \cdot 15^{x} = 5 \cdot (3 \cdot 5)^{x} = 5 \cdot 3^{x} \cdot 5^{x}$

$3 \cdot 5^{x+1} = 3 \cdot 5 \cdot 5^{x} = 3 \cdot 5^{x+1}$

Now the equation becomes:

$5 \cdot 3^{x} \cdot 5^{x} - 3 \cdot 5^{x+1} - 3^{x} + 3 = 0$

Step 2: Combine the like terms. $5 \cdot 3^{x} \cdot 5^{x} - 3 \cdot 5^{x+1} = 5^{x} \cdot 3^{x} - 3 \cdot 5^{x+1}$

Step 3: Factor out common terms. Factor out $5^{x}$ from the first two terms:

$5^{x} \cdot (3^{x} - 3) - 3^{x} + 3 = 0$

Step 4: Combine the terms. Now the equation becomes:

$5^{x} \cdot (3^{x} - 3) - 3^{x} + 3 = 0$

Step 5: Solve for $x$. Let's solve for $x$ by factoring further:

$5^{x} \cdot (3^{x} - 3) - (3^{x} - 3) = 0$

Now, we can factor out $(3^{x} - 3)$ from the expression:

$(3^{x} - 3) \cdot (5^{x} - 1) = 0$

Now set each factor to zero and solve for $x$:

1. $3^{x} - 3 = 0$

Add 3 to both sides:

$3^{x} = 3$

Take the logarithm base 3 of both sides:

$x = \log_{3}(3) = 1$

2. $5^{x} - 1 = 0$

Add 1 to both sides:

$5^{x} = 1$

Since any non-zero number raised to the power of 0 is equal to 1, we know that $x = 0$ satisfies this equation.

So, the solutions for $x$ are $x = 1$ and $x = 0$.