To solve the quadratic equation [tex]\(x^2 + 2 = 20\)[/tex], follow these steps:
1. Rewrite the equation in standard form:
[tex]\[
x^2 + 2 = 20
\][/tex]
Subtract 20 from both sides to get:
[tex]\[
x^2 + 2 - 20 = 0
\][/tex]
Simplifying this, we have:
[tex]\[
x^2 - 18 = 0
\][/tex]
2. Factor the quadratic expression:
Notice that the equation can be written as a difference of squares. The difference of squares formula is [tex]\(a^2 - b^2 = (a + b)(a - b)\)[/tex].
In this case, [tex]\(a = x\)[/tex] and [tex]\(b = \sqrt{18}\)[/tex] which simplifies to [tex]\(3\sqrt{2}\)[/tex]. Therefore:
[tex]\[
x^2 - (3\sqrt{2})^2 = 0 \implies (x - 3\sqrt{2})(x + 3\sqrt{2}) = 0
\][/tex]
3. Set each factor equal to zero:
Now, solve for [tex]\(x\)[/tex] by setting each factor equal to zero:
[tex]\[
x - 3\sqrt{2} = 0 \implies x = 3\sqrt{2}
\][/tex]
[tex]\[
x + 3\sqrt{2} = 0 \implies x = -3\sqrt{2}
\][/tex]
4. Write the solutions:
The solutions to the equation are:
[tex]\[
x = 3\sqrt{2}
\][/tex]
[tex]\[
x = -3\sqrt{2}
\][/tex]
Therefore, the solutions to the equation [tex]\(x^2 + 2 = 20\)[/tex] are [tex]\(x = 3\sqrt{2}\)[/tex] and [tex]\(x = -3\sqrt{2}\)[/tex].
