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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]
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]
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