To solve the equation
[tex]\[ 6x^2 - 2 = 13 \][/tex]
for [tex]\( x \)[/tex], follow these steps:
1. Isolate the quadratic term:
Add 2 to both sides of the equation to move the constant term on the left to the right:
[tex]\[ 6x^2 - 2 + 2 = 13 + 2 \][/tex]
Simplifies to:
[tex]\[ 6x^2 = 15 \][/tex]
2. Solve for [tex]\( x^2 \)[/tex]:
Divide both sides of the equation by 6 to isolate [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{6x^2}{6} = \frac{15}{6} \][/tex]
Simplifies to:
[tex]\[ x^2 = \frac{15}{6} \][/tex]
Further simplify by dividing the numerator and denominator by their greatest common divisor, which is 3:
[tex]\[ x^2 = \frac{5}{2} \][/tex]
3. Solve for [tex]\( x \)[/tex]:
Take the square root of both sides to solve for [tex]\( x \)[/tex]. Remember to include both the positive and negative roots:
[tex]\[ x = \pm \sqrt{\frac{5}{2}} \][/tex]
This can also be written as:
[tex]\[ x = \pm \frac{\sqrt{10}}{2} \][/tex]
Thus, the solutions to the equation [tex]\( 6x^2 - 2 = 13 \)[/tex] are:
[tex]\[ x = \pm \frac{\sqrt{10}}{2} \][/tex]
