You can use the formula for solutions of a quadratic equation to find the solution to the given equation.
The solution to the given equation is given by:
Option B : s = 7
What are the solutions of a quadratic equation?
A quadratic equation is an equation which can be arranged in the standard form [tex]ax^2 + bx + c = 0[/tex] where a, b and c are real numbers and [tex]x[/tex] is variable.
The of [tex]ax^2 + bx + c = 0[/tex] is given by:
[tex]x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
How to find solution(s) of the given equation?
We can see that there is a square root. We can take that root term on one side and can square both the sides, ending up on the quadratic equation.
[tex]s = 4 + \sqrt{s+2}\\s-4 = \sqrt{s+2}\\\\\text{Squaring both sides}\\\\(s-4)^2 = (\sqrt{s+2})^2\\s^2 + 16 - 8s = s+2\\s^2 -9s + 14 = 0[/tex]
You need to notice the most important fact here:
We could've ended up on the same quadratic equation if we start from
[tex]s = 4 - \sqrt{s+2}\\s - 4= -\sqrt{s+2}\\(s-4)^2 = (-\sqrt{s+2})^2\\s^2 + 16 - 8s = s + 2\\s^2 - 9s + 14 = 0[/tex]
This is because [tex](a)^2 = a^2\\and\\(-a)^2 = a^2[/tex]
Thus, from the solutions we will get from the quadratic equation, only one will belong to [tex]s = 4 + \sqrt{s+2}[/tex] since other solution will belong to thee equation [tex]s = 4 - \sqrt{s+2}[/tex]. We will have to check which solution is correct solution to the given equation.
Comparing the obtained quadratic equation with the standard form of quadratic equation, we get:
a = 1 (as there is always 1 multiplied with all terms)
b = -9
c = 14
Thus, the solutions are:
[tex]x = \dfrac{9 \pm \sqrt{81 - 56}}{2}\\\\x = \dfrac{9\pm \sqrt{25}}{2}\\\\x = \dfrac{9+5}{2}, x = \dfrac{9-5}{2}\\\\x = 7, x = 2[/tex]
Checking solutions:
[tex]s = 4 + \sqrt{s+2}\\\text{Putting s = 7}\\7 = 4 + \sqrt{7+2}\\7 = 4 + \sqrt{9}\\7 = 4 + 3 = 7\\\text{Both sides same, thus correct solution}[/tex]
and
[tex]s = 4 + \sqrt{s+2}\\\text{Putting = 2}\\2 = 4 + \sqrt{2+2}\\2 = 4 + \sqrt{4}\\2 = 4 + 2\\2 = 6 \\\text{The above statement is wrong, thus incorrect solution}[/tex]
Thus, s = 7 is the solution to the given equation.
The solution s = 2 is for other equation [tex]s = 4 - \sqrt{s+2}[/tex]
Thus,
The solution to the given equation is given by:
Option B : s = 7
Learn more about quadratic equations here:
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