Solve 11n^2+75n-54

$$11 n ^ { 2 } + 75 n - 54$$

Factor

$$11\left(n-\frac{-3\sqrt{889}-75}{22}\right)\left(n-\frac{3\sqrt{889}-75}{22}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$11n^{2}+75n-54=0$$

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.

$$n=\frac{-75±\sqrt{75^{2}-4\times 11\left(-54\right)}}{2\times 11}$$

Square 75.

$$n=\frac{-75±\sqrt{5625-4\times 11\left(-54\right)}}{2\times 11}$$

Multiply -4 times 11.

$$n=\frac{-75±\sqrt{5625-44\left(-54\right)}}{2\times 11}$$

Multiply -44 times -54.

$$n=\frac{-75±\sqrt{5625+2376}}{2\times 11}$$

Add 5625 to 2376.

$$n=\frac{-75±\sqrt{8001}}{2\times 11}$$

Take the square root of 8001.

$$n=\frac{-75±3\sqrt{889}}{2\times 11}$$

Multiply 2 times 11.

$$n=\frac{-75±3\sqrt{889}}{22}$$

Now solve the equation n=\frac{-75±3\sqrt{889}}{22} when ± is plus. Add -75 to 3\sqrt{889}.

$$n=\frac{3\sqrt{889}-75}{22}$$

Now solve the equation n=\frac{-75±3\sqrt{889}}{22} when ± is minus. Subtract 3\sqrt{889} from -75.

$$n=\frac{-3\sqrt{889}-75}{22}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-75+3\sqrt{889}}{22} for x_{1} and \frac{-75-3\sqrt{889}}{22} for x_{2}.

$$11n^{2}+75n-54=11\left(n-\frac{3\sqrt{889}-75}{22}\right)\left(n-\frac{-3\sqrt{889}-75}{22}\right)$$

Steps Using Direct Factoring Method

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.This is achieved by dividing both sides of the equation by 11

$$x ^ 2 +\frac{75}{11}x -\frac{54}{11} = 0$$

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

$$r + s = -\frac{75}{11} rs = -\frac{54}{11}$$

Two numbers r and s sum up to -\frac{75}{11} exactly when the average of the two numbers is \frac{1}{2}*-\frac{75}{11} = -\frac{75}{22}. 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>

$$r = -\frac{75}{22} - u s = -\frac{75}{22} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{54}{11}

$$(-\frac{75}{22} - u) (-\frac{75}{22} + u) = -\frac{54}{11}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{5625}{484} - u^2 = -\frac{54}{11}$$

Simplify the expression by subtracting \frac{5625}{484} on both sides

$$-u^2 = -\frac{54}{11}-\frac{5625}{484} = -\frac{8001}{484}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{8001}{484} u = \pm\sqrt{\frac{8001}{484}} = \pm \frac{\sqrt{8001}}{22}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{75}{22} - \frac{\sqrt{8001}}{22} = -7.475 s = -\frac{75}{22} + \frac{\sqrt{8001}}{22} = 0.657$$

Evaluate

$$11n^{2}+75n-54$$