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Solve for [tex]$x$[/tex]:

[tex]
2^{x^2 - 53x + 25} = 32^{2 - 9x}
[/tex]

Sagot :

Let's solve the equation step-by-step.

Given:
[tex]\[ 2^{x^2 - 53x + 25} = 32^{2 - 9x} \][/tex]

First, recognize that [tex]\(32\)[/tex] can be written in terms of a base of [tex]\(2\)[/tex] since [tex]\(32 = 2^5\)[/tex]. Thus, the equation can be rewritten:

[tex]\[ 2^{x^2 - 53x + 25} = (2^5)^{2 - 9x} \][/tex]

Next, apply the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:

[tex]\[ 2^{x^2 - 53x + 25} = 2^{5(2 - 9x)} \][/tex]

Since the bases are the same, we can set the exponents equal to each other:

[tex]\[ x^2 - 53x + 25 = 5(2 - 9x) \][/tex]

Expand the right-hand side:

[tex]\[ x^2 - 53x + 25 = 10 - 45x \][/tex]

Move all terms to one side of the equation to set it to zero:

[tex]\[ x^2 - 53x + 25 - 10 + 45x = 0 \][/tex]

Simplify the equation:

[tex]\[ x^2 - 8x + 15 = 0 \][/tex]

This is a quadratic equation. Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a=1\)[/tex], [tex]\(b=-8\)[/tex], and [tex]\(c=15\)[/tex]:

[tex]\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1} \][/tex]

Calculate the discriminant:

[tex]\[ x = \frac{8 \pm \sqrt{64 - 60}}{2} \][/tex]

[tex]\[ x = \frac{8 \pm \sqrt{4}}{2} \][/tex]

[tex]\[ x = \frac{8 \pm 2}{2} \][/tex]

This gives us two solutions:

[tex]\[ x = \frac{8 + 2}{2} = 5 \quad \text{and} \quad x = \frac{8 - 2}{2} = 3 \][/tex]

Thus, the solutions to the equation are:

[tex]\[ \boxed{3 \text{ and } 5} \][/tex]