The positive root of the equation [tex]$x^2+11=30$[/tex] lies between

A. 1 and 3
B. 6 and 7
C. 4 and 5
D. 5 and 6
E. 9 and 10



Answer :

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.