Solve for n
n=2\sqrt{2}-1\approx 1.828427125
n=-2\sqrt{2}-1\approx -3.828427125
Share
Copied to clipboard
n^{2}+2n-1=6
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^{2}+2n-1-6=6-6
Subtract 6 from both sides of the equation.
n^{2}+2n-1-6=0
Subtracting 6 from itself leaves 0.
n^{2}+2n-7=0
Subtract 6 from -1.
n=\frac{-2±\sqrt{2^{2}-4\left(-7\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-2±\sqrt{4-4\left(-7\right)}}{2}
Square 2.
n=\frac{-2±\sqrt{4+28}}{2}
Multiply -4 times -7.
n=\frac{-2±\sqrt{32}}{2}
Add 4 to 28.
n=\frac{-2±4\sqrt{2}}{2}
Take the square root of 32.
n=\frac{4\sqrt{2}-2}{2}
Now solve the equation n=\frac{-2±4\sqrt{2}}{2} when ± is plus. Add -2 to 4\sqrt{2}.
n=2\sqrt{2}-1
Divide 4\sqrt{2}-2 by 2.
n=\frac{-4\sqrt{2}-2}{2}
Now solve the equation n=\frac{-2±4\sqrt{2}}{2} when ± is minus. Subtract 4\sqrt{2} from -2.
n=-2\sqrt{2}-1
Divide -2-4\sqrt{2} by 2.
n=2\sqrt{2}-1 n=-2\sqrt{2}-1
The equation is now solved.
n^{2}+2n-1=6
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}+2n-1-\left(-1\right)=6-\left(-1\right)
Add 1 to both sides of the equation.
n^{2}+2n=6-\left(-1\right)
Subtracting -1 from itself leaves 0.
n^{2}+2n=7
Subtract -1 from 6.
n^{2}+2n+1^{2}=7+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}+2n+1=7+1
Square 1.
n^{2}+2n+1=8
Add 7 to 1.
\left(n+1\right)^{2}=8
Factor n^{2}+2n+1. 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+1\right)^{2}}=\sqrt{8}
Take the square root of both sides of the equation.
n+1=2\sqrt{2} n+1=-2\sqrt{2}
Simplify.
n=2\sqrt{2}-1 n=-2\sqrt{2}-1
Subtract 1 from both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}