https://file.aiquickdraw.com/mathbot/file/17270962493622h5qnej5.jpg

The equation presented in the image is:

\[
\left(\frac{1}{2}\right)^{x^2 + 3x - 7} = \left(\frac{1}{2}\right)^{2x + 1}
\]

To solve this equation, we observe that the bases on both sides are the same, which is \(\frac{1}{2}\). When the bases are equal, we can simply equate the exponents:

\[
x^2 + 3x - 7 = 2x + 1
\]

Now, let's solve this quadratic equation:

1. Rearrange all terms to one side of the equation:

\[
x^2 + 3x - 7 - 2x - 1 = 0
\]

Simplifying:

\[
x^2 + (3x - 2x) + (-7 - 1) = 0
\]

\[
x^2 + x - 8 = 0
\]

2. Now, we solve the quadratic equation \(x^2 + x - 8 = 0\). We can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 + x - 8 = 0\), the coefficients are:
- \(a = 1\),
- \(b = 1\),
- \(c = -8\).

Substituting these into the quadratic formula:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-8)}}{2(1)}
\]

\[
x = \frac{-1 \pm \sqrt{1 + 32}}{2}
\]

\[
x = \frac{-1 \pm \sqrt{33}}{2}
\]

Thus, the solutions are:

\[
x = \frac{-1 + \sqrt{33}}{2} \quad \text{or} \quad x = \frac{-1 - \sqrt{33}}{2}
\]

These are the two solutions for \(x\).

Would you like further details on any part of this solution?

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Here are 5 related questions for practice:
1. How do you solve equations when both sides have the same base but different exponents?
2. Can all quadratic equations be solved using the quadratic formula?
3. How can you factor simple quadratic equations?
4. What are the conditions for an equation to have real or complex solutions?
5. What is the discriminant and how does it affect the type of solutions?

**Tip:** When solving exponential equations with the same base, always focus on equating the exponents directly after verifying that the bases are indeed equal.

Tentukan himpunan penyelesaian dari (1/2)^(x^2 + 3x - 7) = (1/2)^(2x + 1)

Learn how to solve the exponential equation (1/2)^(x^2 + 3x - 7) = (1/2)^(2x + 1) by equating exponents and applying the quadratic formula. This problem covers algebraic concepts such as exponential equations and quadratic equations, suitable for students in Grades 9-12.

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