Let's find the real solution to the radical equation [tex]\( \sqrt{7m + 18} = m \)[/tex].
1. Isolate the Radical:
The given equation is:
[tex]\[
\sqrt{7m + 18} = m
\][/tex]
2. Square Both Sides:
To eliminate the square root, we square both sides of the equation:
[tex]\[
(\sqrt{7m + 18})^2 = m^2
\][/tex]
Simplifying this, we get:
[tex]\[
7m + 18 = m^2
\][/tex]
3. Rearrange into a Standard Quadratic Form:
Move all terms to one side of the equation to set it to zero:
[tex]\[
m^2 - 7m - 18 = 0
\][/tex]
4. Solve the Quadratic Equation:
To find the values of [tex]\( m \)[/tex] that satisfy this quadratic equation, we can use the quadratic formula [tex]\( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = -18 \)[/tex].
Let's calculate the discriminant first:
[tex]\[
\Delta = b^2 - 4ac = (-7)^2 - 4(1)(-18) = 49 + 72 = 121
\][/tex]
Now substitute back into the quadratic formula:
[tex]\[
m = \frac{-(-7) \pm \sqrt{121}}{2(1)} = \frac{7 \pm 11}{2}
\][/tex]
5. Calculate the Roots:
From the quadratic formula, we get two solutions:
[tex]\[
m_1 = \frac{7 + 11}{2} = \frac{18}{2} = 9
\][/tex]
[tex]\[
m_2 = \frac{7 - 11}{2} = \frac{-4}{2} = -2
\][/tex]
6. Check for Real Solution:
We need to verify which of these solutions are valid for the original equation. A square root function must yield a non-negative value since the output of a square root should be non-negative.
Plugging [tex]\( m = 9 \)[/tex] back into the original equation, we get:
[tex]\[
\sqrt{7(9) + 18} = \sqrt{63 + 18} = \sqrt{81} = 9
\][/tex]
So, [tex]\( m = 9 \)[/tex] is a valid solution.
Plugging [tex]\( m = -2 \)[/tex] back into the original equation, we get:
[tex]\[
\sqrt{7(-2) + 18} = \sqrt{-14 + 18} = \sqrt{4} = 2
\][/tex]
However, the left side gives 2, which does not equal [tex]\(-2\)[/tex] on the right side. So, [tex]\( m = -2 \)[/tex] is not a valid solution.
Thus, the only real solution to the equation [tex]\(\sqrt{7m + 18} = m\)[/tex] is:
[tex]\[
m = 9
\][/tex]
