To solve the quadratic equation [tex]\( x^2 + 2x + 7 = 21 \)[/tex]:
1. Rearrange the equation: Move all terms to one side to set the equation to zero.
[tex]\[
x^2 + 2x + 7 - 21 = 0 \rightarrow x^2 + 2x - 14 = 0
\][/tex]
2. Solve the quadratic equation using the quadratic formula [tex]\( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -14 \)[/tex]:
[tex]\[
x = \frac{{-2 \pm \sqrt{{2^2 - 4 \cdot 1 \cdot (-14)}}}}{2 \cdot 1} = \frac{{-2 \pm \sqrt{{4 + 56}}}}{2} = \frac{{-2 \pm \sqrt{{60}}}}{2}
\][/tex]
3. Simplify the solutions:
[tex]\[
x = \frac{{-2 \pm \sqrt{60}}}{2} = \frac{{-2 \pm 2\sqrt{15}}}{2} = -1 \pm \sqrt{15}
\][/tex]
4. Identify the positive solution: Among the solutions [tex]\( x = -1 + \sqrt{15} \)[/tex] and [tex]\( x = -1 - \sqrt{15} \)[/tex], only [tex]\( x = -1 + \sqrt{15} \)[/tex] is positive.
5. Count the number of positive solutions: There is only 1 positive solution.
6. Approximate the greatest positive solution: Calculate and approximate to the nearest hundredth.
[tex]\[
-1 + \sqrt{15} \approx -1 + 3.872 = 2.872
\][/tex]
Rounded to the nearest hundredth, the value is 2.87.
Therefore:
- The number of positive solutions is [tex]\( \boxed{1} \)[/tex].
- The approximate value of the greatest solution to the equation, rounded to the nearest hundredth, is [tex]\( \boxed{2.87} \)[/tex].
