Solve for [tex]\( x \)[/tex]:

[tex]\[ 9x^2 + 17 = 1493 \][/tex]

A. [tex]\(\left\{\frac{i \sqrt{785}}{5},-\frac{i \sqrt{785}}{5}\right\}\)[/tex]

B. [tex]\(\{2 \sqrt{41},-2 \sqrt{41}\}\)[/tex]

C. [tex]\(\left\{\frac{\sqrt{1510}}{3},-\frac{\sqrt{1510}}{3}\right\}\)[/tex]

D. [tex]\(\left\{\frac{i \sqrt{935}}{5}\right\}\)[/tex]



Answer :

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]