To solve the quadratic equation [tex]\(x^2 + 2x + 7 = 21\)[/tex], follow these steps:
1. Move all terms to one side of the equation:
[tex]\[x^2 + 2x + 7 - 21 = 0\][/tex]
[tex]\[x^2 + 2x - 14 = 0\][/tex]
2. Use the quadratic formula to solve for [tex]\(x\)[/tex]. The quadratic formula is:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = -14\)[/tex].
3. Calculate the discriminant ([tex]\(b^2 - 4ac\)[/tex]):
[tex]\[
b^2 - 4ac = 2^2 - 4(1)(-14) = 4 + 56 = 60
\][/tex]
4. Substitute the values into the quadratic formula:
[tex]\[
x = \frac{-2 \pm \sqrt{60}}{2}
\][/tex]
5. Simplify the expression under the square root and the fraction:
[tex]\[
\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}
\][/tex]
[tex]\[
x = \frac{-2 \pm 2\sqrt{15}}{2}
\][/tex]
[tex]\[
x = -1 \pm \sqrt{15}
\][/tex]
So, the solutions are [tex]\(x = -1 + \sqrt{15}\)[/tex] and [tex]\(x = -1 - \sqrt{15}\)[/tex].
6. Identify the number of positive solutions:
[tex]\[
-1 + \sqrt{15} \quad \text{(positive if} \sqrt{15} > 1\text{)}
\][/tex]
[tex]\[
-1 - \sqrt{15} \quad \text{(always negative)}
\][/tex]
Since [tex]\(\sqrt{15}\)[/tex] is approximately 3.87, [tex]\( -1 + \sqrt{15} \)[/tex] is positive and [tex]\( -1 - \sqrt{15} \)[/tex] is negative.
There is 1 positive solution.
7. Determine the approximate value of the greatest solution:
[tex]\[
-1 + \sqrt{15} \approx -1 + 3.87 \approx 2.87
\][/tex]
Therefore, the answers are:
- The number of positive solutions to this equation 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].
