IDNLearn.com is your go-to resource for finding answers to any question you have. Discover prompt and accurate responses from our experts, ensuring you get the information you need quickly.
Sagot :
To solve the equation:
[tex]\[3^{x^2 - 8x + 15} = 81^{2 - 4x}\][/tex]
we need to use some properties of exponents and logarithms for solving it step by step.
1. Notice that [tex]\(81 = 3^4\)[/tex]. Therefore, we can rewrite [tex]\(81^{2 - 4x}\)[/tex] as:
[tex]\[ 81^{2 - 4x} = (3^4)^{2 - 4x} = 3^{4(2 - 4x)} = 3^{8 - 16x} \][/tex]
2. Now, we have the equation:
[tex]\[ 3^{x^2 - 8x + 15} = 3^{8 - 16x} \][/tex]
Because the bases (3) are the same, we can equate the exponents:
[tex]\[ x^2 - 8x + 15 = 8 - 16x \][/tex]
3. Simplify the equation:
[tex]\[ x^2 - 8x + 15 = 8 - 16x \][/tex]
Move all terms to one side to set the equation to zero:
[tex]\[ x^2 - 8x + 15 - 8 + 16x = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 + 8x + 7 = 0 \][/tex]
4. Factor the quadratic equation:
[tex]\[ x^2 + 8x + 7 = (x + 7)(x + 1) = 0 \][/tex]
5. Set each factor equal to zero:
[tex]\[ x + 7 = 0 \quad \text{or} \quad x + 1 = 0 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ x = -7 \quad \text{or} \quad x = -1 \][/tex]
Thus, the solutions to the equation are:
[tex]\[ x = -7, -1 \][/tex]
[tex]\[3^{x^2 - 8x + 15} = 81^{2 - 4x}\][/tex]
we need to use some properties of exponents and logarithms for solving it step by step.
1. Notice that [tex]\(81 = 3^4\)[/tex]. Therefore, we can rewrite [tex]\(81^{2 - 4x}\)[/tex] as:
[tex]\[ 81^{2 - 4x} = (3^4)^{2 - 4x} = 3^{4(2 - 4x)} = 3^{8 - 16x} \][/tex]
2. Now, we have the equation:
[tex]\[ 3^{x^2 - 8x + 15} = 3^{8 - 16x} \][/tex]
Because the bases (3) are the same, we can equate the exponents:
[tex]\[ x^2 - 8x + 15 = 8 - 16x \][/tex]
3. Simplify the equation:
[tex]\[ x^2 - 8x + 15 = 8 - 16x \][/tex]
Move all terms to one side to set the equation to zero:
[tex]\[ x^2 - 8x + 15 - 8 + 16x = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 + 8x + 7 = 0 \][/tex]
4. Factor the quadratic equation:
[tex]\[ x^2 + 8x + 7 = (x + 7)(x + 1) = 0 \][/tex]
5. Set each factor equal to zero:
[tex]\[ x + 7 = 0 \quad \text{or} \quad x + 1 = 0 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ x = -7 \quad \text{or} \quad x = -1 \][/tex]
Thus, the solutions to the equation are:
[tex]\[ x = -7, -1 \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.