To solve the quadratic equation [tex]\( 5x^2 - 15 = 0 \)[/tex], follow these detailed steps:
1. Isolate the term with [tex]\( x^2 \)[/tex]:
- Start by adding 15 to both sides of the equation to move the constant term to the right side:
[tex]\[
5x^2 - 15 + 15 = 0 + 15
\][/tex]
Simplifying both sides, we get:
[tex]\[
5x^2 = 15
\][/tex]
2. Solve for [tex]\( x^2 \)[/tex]:
- Divide both sides by 5 to isolate [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{5x^2}{5} = \frac{15}{5}
\][/tex]
Simplifying both sides, we get:
[tex]\[
x^2 = 3
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
- Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{3}
\][/tex]
4. Simplify the square root:
- The two possible solutions are the positive and negative square roots of 3:
[tex]\[
x_1 = \sqrt{3} \approx 1.7320508075688772
\][/tex]
[tex]\[
x_2 = -\sqrt{3} \approx -1.7320508075688772
\][/tex]
So, the two possible solutions to the equation [tex]\( 5x^2 - 15 = 0 \)[/tex] are:
[tex]\( 1.7320508075688772, -1.7320508075688772 \)[/tex]
