To solve the equation [tex]\(9x^2 + 17 = 1493\)[/tex], we'll go through the following steps:
1. Rewrite the equation in standard form:
[tex]\[
9x^2 + 17 = 1493
\][/tex]
Subtract 1493 from both sides to get:
[tex]\[
9x^2 + 17 - 1493 = 0
\][/tex]
Simplify the constants on the left-hand side:
[tex]\[
9x^2 - 1476 = 0
\][/tex]
2. Isolate the quadratic term:
[tex]\[
9x^2 = 1476
\][/tex]
3. Solve for [tex]\( x^2 \)[/tex]:
Divide both sides of the equation by 9:
[tex]\[
x^2 = \frac{1476}{9}
\][/tex]
Simplify the fraction:
[tex]\[
x^2 = 164
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Take the square root of both sides:
[tex]\[
x = \pm \sqrt{164}
\][/tex]
5. Simplify the square root:
Notice that [tex]\(164\)[/tex] can be factored into [tex]\(4 \times 41\)[/tex]:
[tex]\[
\sqrt{164} = \sqrt{4 \times 41} = \sqrt{4} \times \sqrt{41} = 2\sqrt{41}
\][/tex]
So, the solutions to the equation are:
[tex]\[
x = -2\sqrt{41} \quad \text{and} \quad x = 2\sqrt{41}
\][/tex]
Therefore, the correct solutions are:
[tex]\[
\{2\sqrt{41}, -2\sqrt{41}\}
\][/tex]
