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m^{2}-8m+1=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
m=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-\left(-8\right)±\sqrt{64-4}}{2}
Square -8.
m=\frac{-\left(-8\right)±\sqrt{60}}{2}
Add 64 to -4.
m=\frac{-\left(-8\right)±2\sqrt{15}}{2}
Take the square root of 60.
m=\frac{8±2\sqrt{15}}{2}
The opposite of -8 is 8.
m=\frac{2\sqrt{15}+8}{2}
Now solve the equation m=\frac{8±2\sqrt{15}}{2} when ± is plus. Add 8 to 2\sqrt{15}.
m=\sqrt{15}+4
Divide 8+2\sqrt{15} by 2.
m=\frac{8-2\sqrt{15}}{2}
Now solve the equation m=\frac{8±2\sqrt{15}}{2} when ± is minus. Subtract 2\sqrt{15} from 8.
m=4-\sqrt{15}
Divide 8-2\sqrt{15} by 2.
m=\sqrt{15}+4 m=4-\sqrt{15}
The equation is now solved.
m^{2}-8m+1=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
m^{2}-8m+1-1=-1
Subtract 1 from both sides of the equation.
m^{2}-8m=-1
Subtracting 1 from itself leaves 0.
m^{2}-8m+\left(-4\right)^{2}=-1+\left(-4\right)^{2}
Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
m^{2}-8m+16=-1+16
Square -4.
m^{2}-8m+16=15
Add -1 to 16.
\left(m-4\right)^{2}=15
Factor m^{2}-8m+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(m-4\right)^{2}}=\sqrt{15}
Take the square root of both sides of the equation.
m-4=\sqrt{15} m-4=-\sqrt{15}
Simplify.
m=\sqrt{15}+4 m=4-\sqrt{15}
Add 4 to both sides of the equation.