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Solve for n (complex solution)
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n^{2}+8n-6=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.
n=\frac{-8±\sqrt{8^{2}-4\left(-6\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-8±\sqrt{64-4\left(-6\right)}}{2}
Square 8.
n=\frac{-8±\sqrt{64+24}}{2}
Multiply -4 times -6.
n=\frac{-8±\sqrt{88}}{2}
Add 64 to 24.
n=\frac{-8±2\sqrt{22}}{2}
Take the square root of 88.
n=\frac{2\sqrt{22}-8}{2}
Now solve the equation n=\frac{-8±2\sqrt{22}}{2} when ± is plus. Add -8 to 2\sqrt{22}.
n=\sqrt{22}-4
Divide -8+2\sqrt{22} by 2.
n=\frac{-2\sqrt{22}-8}{2}
Now solve the equation n=\frac{-8±2\sqrt{22}}{2} when ± is minus. Subtract 2\sqrt{22} from -8.
n=-\sqrt{22}-4
Divide -8-2\sqrt{22} by 2.
n=\sqrt{22}-4 n=-\sqrt{22}-4
The equation is now solved.
n^{2}+8n-6=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.
n^{2}+8n-6-\left(-6\right)=-\left(-6\right)
Add 6 to both sides of the equation.
n^{2}+8n=-\left(-6\right)
Subtracting -6 from itself leaves 0.
n^{2}+8n=6
Subtract -6 from 0.
n^{2}+8n+4^{2}=6+4^{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.
n^{2}+8n+16=6+16
Square 4.
n^{2}+8n+16=22
Add 6 to 16.
\left(n+4\right)^{2}=22
Factor n^{2}+8n+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(n+4\right)^{2}}=\sqrt{22}
Take the square root of both sides of the equation.
n+4=\sqrt{22} n+4=-\sqrt{22}
Simplify.
n=\sqrt{22}-4 n=-\sqrt{22}-4
Subtract 4 from both sides of the equation.
x ^ 2 +8x -6 = 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 = -8 rs = -6
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 = -4 - u s = -4 + u
Two numbers r and s sum up to -8 exactly when the average of the two numbers is \frac{1}{2}*-8 = -4. 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>
(-4 - u) (-4 + u) = -6
To solve for unknown quantity u, substitute these in the product equation rs = -6
16 - u^2 = -6
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -6-16 = -22
Simplify the expression by subtracting 16 on both sides
u^2 = 22 u = \pm\sqrt{22} = \pm \sqrt{22}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-4 - \sqrt{22} = -8.690 s = -4 + \sqrt{22} = 0.690
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
n^{2}+8n-6=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.
n=\frac{-8±\sqrt{8^{2}-4\left(-6\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-8±\sqrt{64-4\left(-6\right)}}{2}
Square 8.
n=\frac{-8±\sqrt{64+24}}{2}
Multiply -4 times -6.
n=\frac{-8±\sqrt{88}}{2}
Add 64 to 24.
n=\frac{-8±2\sqrt{22}}{2}
Take the square root of 88.
n=\frac{2\sqrt{22}-8}{2}
Now solve the equation n=\frac{-8±2\sqrt{22}}{2} when ± is plus. Add -8 to 2\sqrt{22}.
n=\sqrt{22}-4
Divide -8+2\sqrt{22} by 2.
n=\frac{-2\sqrt{22}-8}{2}
Now solve the equation n=\frac{-8±2\sqrt{22}}{2} when ± is minus. Subtract 2\sqrt{22} from -8.
n=-\sqrt{22}-4
Divide -8-2\sqrt{22} by 2.
n=\sqrt{22}-4 n=-\sqrt{22}-4
The equation is now solved.
n^{2}+8n-6=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.
n^{2}+8n-6-\left(-6\right)=-\left(-6\right)
Add 6 to both sides of the equation.
n^{2}+8n=-\left(-6\right)
Subtracting -6 from itself leaves 0.
n^{2}+8n=6
Subtract -6 from 0.
n^{2}+8n+4^{2}=6+4^{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.
n^{2}+8n+16=6+16
Square 4.
n^{2}+8n+16=22
Add 6 to 16.
\left(n+4\right)^{2}=22
Factor n^{2}+8n+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(n+4\right)^{2}}=\sqrt{22}
Take the square root of both sides of the equation.
n+4=\sqrt{22} n+4=-\sqrt{22}
Simplify.
n=\sqrt{22}-4 n=-\sqrt{22}-4
Subtract 4 from both sides of the equation.