Solve for n
n = \frac{\sqrt{41} + 7}{2} \approx 6.701562119
n=\frac{7-\sqrt{41}}{2}\approx 0.298437881
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n^{2}-7n+2=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{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 2}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-\left(-7\right)±\sqrt{49-4\times 2}}{2}
Square -7.
n=\frac{-\left(-7\right)±\sqrt{49-8}}{2}
Multiply -4 times 2.
n=\frac{-\left(-7\right)±\sqrt{41}}{2}
Add 49 to -8.
n=\frac{7±\sqrt{41}}{2}
The opposite of -7 is 7.
n=\frac{\sqrt{41}+7}{2}
Now solve the equation n=\frac{7±\sqrt{41}}{2} when ± is plus. Add 7 to \sqrt{41}.
n=\frac{7-\sqrt{41}}{2}
Now solve the equation n=\frac{7±\sqrt{41}}{2} when ± is minus. Subtract \sqrt{41} from 7.
n=\frac{\sqrt{41}+7}{2} n=\frac{7-\sqrt{41}}{2}
The equation is now solved.
n^{2}-7n+2=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}-7n+2-2=-2
Subtract 2 from both sides of the equation.
n^{2}-7n=-2
Subtracting 2 from itself leaves 0.
n^{2}-7n+\left(-\frac{7}{2}\right)^{2}=-2+\left(-\frac{7}{2}\right)^{2}
Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}-7n+\frac{49}{4}=-2+\frac{49}{4}
Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.
n^{2}-7n+\frac{49}{4}=\frac{41}{4}
Add -2 to \frac{49}{4}.
\left(n-\frac{7}{2}\right)^{2}=\frac{41}{4}
Factor n^{2}-7n+\frac{49}{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(n-\frac{7}{2}\right)^{2}}=\sqrt{\frac{41}{4}}
Take the square root of both sides of the equation.
n-\frac{7}{2}=\frac{\sqrt{41}}{2} n-\frac{7}{2}=-\frac{\sqrt{41}}{2}
Simplify.
n=\frac{\sqrt{41}+7}{2} n=\frac{7-\sqrt{41}}{2}
Add \frac{7}{2} to both sides of the equation.
x ^ 2 -7x +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 = 7 rs = 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{7}{2} - u s = \frac{7}{2} + u
Two numbers r and s sum up to 7 exactly when the average of the two numbers is \frac{1}{2}*7 = \frac{7}{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{7}{2} - u) (\frac{7}{2} + u) = 2
To solve for unknown quantity u, substitute these in the product equation rs = 2
\frac{49}{4} - u^2 = 2
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 2-\frac{49}{4} = -\frac{41}{4}
Simplify the expression by subtracting \frac{49}{4} on both sides
u^2 = \frac{41}{4} u = \pm\sqrt{\frac{41}{4}} = \pm \frac{\sqrt{41}}{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{7}{2} - \frac{\sqrt{41}}{2} = 0.298 s = \frac{7}{2} + \frac{\sqrt{41}}{2} = 6.702
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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