To determine the interval in which the positive root of the equation [tex]\( x^2 + 11 = 30 \)[/tex] lies, let's proceed with the following steps:
1. Simplify the Equation:
[tex]\[
x^2 + 11 = 30
\][/tex]
Subtract 11 from both sides:
[tex]\[
x^2 = 19
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
Take the square root of both sides:
[tex]\[
x = \pm \sqrt{19}
\][/tex]
The solutions are [tex]\( x = \sqrt{19} \)[/tex] and [tex]\( x = -\sqrt{19} \)[/tex]. Since we are interested in the positive root, we consider [tex]\( x = \sqrt{19} \)[/tex].
3. Approximate [tex]\( \sqrt{19} \)[/tex]:
To determine the interval in which [tex]\( \sqrt{19} \)[/tex] lies, we need an approximate value. The square root of 19 is approximately 4.36.
4. Compare with Intervals:
Now we need to check which of the given intervals contains approximately 4.36. The intervals are:
- (1, 3)
- (6, 7)
- (4, 5)
- (5, 6)
- (9, 10)
Comparing [tex]\( \sqrt{19} \approx 4.36 \)[/tex] with the intervals:
[tex]\[
1 < 4.36 < 3 \quad (\text{False})
\][/tex]
[tex]\[
6 < 4.36 < 7 \quad (\text{False})
\][/tex]
[tex]\[
4 < 4.36 < 5 \quad (\text{True})
\][/tex]
[tex]\[
5 < 4.36 < 6 \quad (\text{False})
\][/tex]
[tex]\[
9 < 4.36 < 10 \quad (\text{False})
\][/tex]
Thus, the interval that contains [tex]\( \sqrt{19} \)[/tex] is [tex]\((4, 5)\)[/tex].
Therefore, the positive root of the equation [tex]\( x^2 + 11 = 30 \)[/tex] lies between 4 and 5.
