Solve 16n^2-148n+612

$$16 n ^ { 2 } - 148 n + 612$$

Factor

$$4\left(4n^{2}-37n+153\right)$$

Solution Steps

Factor out 4. Polynomial 4n^{2}-37n+153 is not factored since it does not have any rational roots.

$$4\left(4n^{2}-37n+153\right)$$

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.

$$16n^{2}-148n+612=0$$

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.

$$n=\frac{-\left(-148\right)±\sqrt{\left(-148\right)^{2}-4\times 16\times 612}}{2\times 16}$$

Square -148.

$$n=\frac{-\left(-148\right)±\sqrt{21904-4\times 16\times 612}}{2\times 16}$$

Multiply -4 times 16.

$$n=\frac{-\left(-148\right)±\sqrt{21904-64\times 612}}{2\times 16}$$

Multiply -64 times 612.

$$n=\frac{-\left(-148\right)±\sqrt{21904-39168}}{2\times 16}$$

Add 21904 to -39168.

$$n=\frac{-\left(-148\right)±\sqrt{-17264}}{2\times 16}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

$$16n^{2}-148n+612$$

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 16

$$x ^ 2 -\frac{37}{4}x +\frac{153}{4} = 0$$

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

$$r + s = \frac{37}{4} rs = \frac{153}{4}$$

Two numbers r and s sum up to \frac{37}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{37}{4} = \frac{37}{8}. 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>

$$r = \frac{37}{8} - u s = \frac{37}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{153}{4}

$$(\frac{37}{8} - u) (\frac{37}{8} + u) = \frac{153}{4}$$

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

$$\frac{1369}{64} - u^2 = \frac{153}{4}$$

Simplify the expression by subtracting \frac{1369}{64} on both sides

$$-u^2 = \frac{153}{4}-\frac{1369}{64} = \frac{1079}{64}$$

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

$$u^2 = -\frac{1079}{64} u = \pm\sqrt{-\frac{1079}{64}} = \pm \frac{\sqrt{1079}}{8}i$$

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

$$r =\frac{37}{8} - \frac{\sqrt{1079}}{8}i = 4.625 - 4.106i s = \frac{37}{8} + \frac{\sqrt{1079}}{8}i = 4.625 + 4.106i$$

Evaluate

$$16n^{2}-148n+612$$