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-8m^{2}+8m+4=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{-8±\sqrt{8^{2}-4\left(-8\right)\times 4}}{2\left(-8\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 8 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-8±\sqrt{64-4\left(-8\right)\times 4}}{2\left(-8\right)}
Square 8.
m=\frac{-8±\sqrt{64+32\times 4}}{2\left(-8\right)}
Multiply -4 times -8.
m=\frac{-8±\sqrt{64+128}}{2\left(-8\right)}
Multiply 32 times 4.
m=\frac{-8±\sqrt{192}}{2\left(-8\right)}
Add 64 to 128.
m=\frac{-8±8\sqrt{3}}{2\left(-8\right)}
Take the square root of 192.
m=\frac{-8±8\sqrt{3}}{-16}
Multiply 2 times -8.
m=\frac{8\sqrt{3}-8}{-16}
Now solve the equation m=\frac{-8±8\sqrt{3}}{-16} when ± is plus. Add -8 to 8\sqrt{3}.
m=\frac{1-\sqrt{3}}{2}
Divide -8+8\sqrt{3} by -16.
m=\frac{-8\sqrt{3}-8}{-16}
Now solve the equation m=\frac{-8±8\sqrt{3}}{-16} when ± is minus. Subtract 8\sqrt{3} from -8.
m=\frac{\sqrt{3}+1}{2}
Divide -8-8\sqrt{3} by -16.
m=\frac{1-\sqrt{3}}{2} m=\frac{\sqrt{3}+1}{2}
The equation is now solved.
-8m^{2}+8m+4=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.
-8m^{2}+8m+4-4=-4
Subtract 4 from both sides of the equation.
-8m^{2}+8m=-4
Subtracting 4 from itself leaves 0.
\frac{-8m^{2}+8m}{-8}=-\frac{4}{-8}
Divide both sides by -8.
m^{2}+\frac{8}{-8}m=-\frac{4}{-8}
Dividing by -8 undoes the multiplication by -8.
m^{2}-m=-\frac{4}{-8}
Divide 8 by -8.
m^{2}-m=\frac{1}{2}
Reduce the fraction \frac{-4}{-8} to lowest terms by extracting and canceling out 4.
m^{2}-m+\left(-\frac{1}{2}\right)^{2}=\frac{1}{2}+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
m^{2}-m+\frac{1}{4}=\frac{1}{2}+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
m^{2}-m+\frac{1}{4}=\frac{3}{4}
Add \frac{1}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(m-\frac{1}{2}\right)^{2}=\frac{3}{4}
Factor m^{2}-m+\frac{1}{4}. 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-\frac{1}{2}\right)^{2}}=\sqrt{\frac{3}{4}}
Take the square root of both sides of the equation.
m-\frac{1}{2}=\frac{\sqrt{3}}{2} m-\frac{1}{2}=-\frac{\sqrt{3}}{2}
Simplify.
m=\frac{\sqrt{3}+1}{2} m=\frac{1-\sqrt{3}}{2}
Add \frac{1}{2} to both sides of the equation.
x ^ 2 -1x -\frac{1}{2} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.
r + s = 1 rs = -\frac{1}{2}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = \frac{1}{2} - u s = \frac{1}{2} + u
Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(\frac{1}{2} - u) (\frac{1}{2} + u) = -\frac{1}{2}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{2}
\frac{1}{4} - u^2 = -\frac{1}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{1}{2}-\frac{1}{4} = -\frac{3}{4}
Simplify the expression by subtracting \frac{1}{4} on both sides
u^2 = \frac{3}{4} u = \pm\sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{1}{2} - \frac{\sqrt{3}}{2} = -0.366 s = \frac{1}{2} + \frac{\sqrt{3}}{2} = 1.366
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.