To solve the problem of finding the square root of [tex]\(11 + 2 \sqrt{30}\)[/tex], we'll proceed with the following steps:
1. Assume the form of the square root:
Let [tex]\(x\)[/tex] be the square root we are looking for. Assume that
[tex]\[
x = \sqrt{a} + \sqrt{b}
\][/tex]
for some values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
2. Square both sides:
Squaring both sides of the equation [tex]\(x = \sqrt{a} + \sqrt{b}\)[/tex] gives
[tex]\[
x^2 = (\sqrt{a} + \sqrt{b})^2 = a + b + 2 \sqrt{ab}
\][/tex]
3. Setup the equation:
We know that
[tex]\[
x^2 = 11 + 2 \sqrt{30}
\][/tex]
Thus, from the equation [tex]\(x^2 = a + b + 2 \sqrt{ab}\)[/tex], we get
[tex]\[
a + b + 2 \sqrt{ab} = 11 + 2 \sqrt{30}
\][/tex]
4. Equate the rational and irrational parts:
By comparing both sides, we can separate the rational and irrational parts:
[tex]\[
a + b = 11
\][/tex]
and
[tex]\[
2 \sqrt{ab} = 2 \sqrt{30}
\][/tex]
5. Solve for [tex]\(ab\)[/tex]:
From the equation [tex]\(2 \sqrt{ab} = 2 \sqrt{30}\)[/tex], we get
[tex]\[
\sqrt{ab} = \sqrt{30}
\][/tex]
Squaring both sides,
[tex]\[
ab = 30
\][/tex]
6. Form and solve the quadratic equation:
Now, we have a system of two equations:
[tex]\[
a + b = 11
\][/tex]
[tex]\[
ab = 30
\][/tex]
These can be thought of as the sum and product of the roots of a quadratic equation. The corresponding quadratic equation is
[tex]\[
t^2 - (a + b)t + ab = 0
\][/tex]
Substituting [tex]\(a + b = 11\)[/tex] and [tex]\(ab = 30\)[/tex],
[tex]\[
t^2 - 11t + 30 = 0
\][/tex]
Solving this quadratic equation using the quadratic formula [tex]\(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], we substitute [tex]\(a = 1\)[/tex], [tex]\(b = -11\)[/tex], and [tex]\(c = 30\)[/tex]:
[tex]\[
t = \frac{11 \pm \sqrt{11^2 - 4 \cdot 1 \cdot 30}}{2 \cdot 1}
\][/tex]
[tex]\[
t = \frac{11 \pm \sqrt{121 - 120}}{2}
\][/tex]
[tex]\[
t = \frac{11 \pm \sqrt{1}}{2}
\][/tex]
[tex]\[
t = \frac{11 \pm 1}{2}
\][/tex]
Therefore, we get two solutions:
[tex]\[
t_1 = \frac{12}{2} = 6
\][/tex]
[tex]\[
t_2 = \frac{10}{2} = 5
\][/tex]
Hence, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are 6 and 5, respectively.
7. Construct the square root:
Therefore, the square root of [tex]\(11 + 2 \sqrt{30}\)[/tex] is
[tex]\[
\sqrt{11 + 2 \sqrt{30}} = \sqrt{6} + \sqrt{5}
\][/tex]
