The quadratic equation $x^2+4mx+m = 2x - 6$ has exactly one real root. Find the positive value of $m$.
We can rewrite the given equation as $x^2 + (4m - 2)x + (m + 6) = 0.$ Since there is only one real root, the discriminant must be 0. Hence,
\[(4m - 2)^2 - 4(m + 6) = 0.\]Expanding, we get $16m^2 - 56m + 4 = 0,$ which factors as $4(4m^2 - 14m + 1) = 0.$ By the quadratic formula,
\[m = \frac{14 \pm \sqrt{14^2 - 4 \cdot 4}}{8} = \frac{14 \pm 6 \sqrt{2}}{8} = \frac{7 \pm 3 \sqrt{2}}{4}.\]The positive solution is $\boxed{\frac{7 + 3 \sqrt{2}}{4}}.$
wrong again.
If it has one root, the discriminant is zero.
x^2+4mx+m = 2x - 6
x^2+(4m-2)x+(m-6) = 0
so we need (4m-2)^2-4(m+6) = 0
(2m-1)^2 - (m+6) = 0
4m^2-5m-5 = 0
m = (5±√105)/8
so the positive value is (5+√105)/8
You are right. I apologize for the mistake. The positive value of $m$ is indeed $\boxed{\frac{5+\sqrt{105}}{8}}.$
To solve this problem, we need to set up the quadratic equation and solve for $m$.
Given the equation $x^2+4mx+m = 2x - 6$, we can rewrite it as $x^2+(4m-2)x+(m+6) = 0$.
For a quadratic equation to have exactly one real root, the discriminant must be equal to zero. The discriminant is calculated as $b^2-4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.
In this case, $a=1$, $b=4m-2$, and $c=m+6$. The discriminant is:
$(4m-2)^2 - 4(1)(m+6) = 0$
Expanding and simplifying, we get:
$16m^2 - 16m - 44 = 0$
To solve this quadratic equation, we can apply the quadratic formula:
$m = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
Plugging in the values, we have:
$m = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(16)(-44)}}{2(16)}$
Simplifying, we get:
$m = \frac{16 \pm \sqrt{256 + 2816}}{32} = \frac{16 \pm \sqrt{3072}}{32}$
$= \frac{16 \pm \sqrt{32 \cdot 96}}{32} = \frac{16 \pm 8\sqrt{3}}{32}$
Factoring out a common factor of 8 from the numerator:
$m = \frac{8(2 \pm \sqrt{3})}{32} = \frac{2 \pm \sqrt{3}}{4}$
Since we are looking for the positive value of $m$, we take the positive value of the expression:
$m = \frac{2 + \sqrt{3}}{4}$
Therefore, the positive value of $m$ is $\boxed{\frac{2 + \sqrt{3}}{4}}$.