Solve 41z^2-984z+5907=0

$$41 z ^ { 2 } - 984 z + 5907 = 0$$

Solve for z

$$z=\frac{\sqrt{123}i}{41}+12\approx 12+0.27050089i$$

$$z=-\frac{\sqrt{123}i}{41}+12\approx 12-0.27050089i$$

Steps Using the Quadratic Formula

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.

$$41z^{2}-984z+5907=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 41 for a, -984 for b, and 5907 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-984\right)±\sqrt{\left(-984\right)^{2}-4\times 41\times 5907}}{2\times 41}$$

Square -984.

$$z=\frac{-\left(-984\right)±\sqrt{968256-4\times 41\times 5907}}{2\times 41}$$

Multiply -4 times 41.

$$z=\frac{-\left(-984\right)±\sqrt{968256-164\times 5907}}{2\times 41}$$

Multiply -164 times 5907.

$$z=\frac{-\left(-984\right)±\sqrt{968256-968748}}{2\times 41}$$

Add 968256 to -968748.

$$z=\frac{-\left(-984\right)±\sqrt{-492}}{2\times 41}$$

Take the square root of -492.

$$z=\frac{-\left(-984\right)±2\sqrt{123}i}{2\times 41}$$

The opposite of -984 is 984.

$$z=\frac{984±2\sqrt{123}i}{2\times 41}$$

Multiply 2 times 41.

$$z=\frac{984±2\sqrt{123}i}{82}$$

Now solve the equation z=\frac{984±2\sqrt{123}i}{82} when ± is plus. Add 984 to 2i\sqrt{123}.

$$z=\frac{984+2\sqrt{123}i}{82}$$

Divide 984+2i\sqrt{123} by 82.

$$z=\frac{\sqrt{123}i}{41}+12$$

Now solve the equation z=\frac{984±2\sqrt{123}i}{82} when ± is minus. Subtract 2i\sqrt{123} from 984.

$$z=\frac{-2\sqrt{123}i+984}{82}$$

Divide 984-2i\sqrt{123} by 82.

$$z=-\frac{\sqrt{123}i}{41}+12$$

The equation is now solved.

$$z=\frac{\sqrt{123}i}{41}+12 z=-\frac{\sqrt{123}i}{41}+12$$

Steps for Completing the Square

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.

$$41z^{2}-984z+5907=0$$

Subtract 5907 from both sides of the equation.

$$41z^{2}-984z+5907-5907=-5907$$

Subtracting 5907 from itself leaves 0.

$$41z^{2}-984z=-5907$$

Divide both sides by 41.

$$\frac{41z^{2}-984z}{41}=-\frac{5907}{41}$$

Dividing by 41 undoes the multiplication by 41.

$$z^{2}+\left(-\frac{984}{41}\right)z=-\frac{5907}{41}$$

Divide -984 by 41.

$$z^{2}-24z=-\frac{5907}{41}$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-24z+\left(-12\right)^{2}=-\frac{5907}{41}+\left(-12\right)^{2}$$

Square -12.

$$z^{2}-24z+144=-\frac{5907}{41}+144$$

Add -\frac{5907}{41} to 144.

$$z^{2}-24z+144=-\frac{3}{41}$$

Factor z^{2}-24z+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-12\right)^{2}=-\frac{3}{41}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-12\right)^{2}}=\sqrt{-\frac{3}{41}}$$

Simplify.

$$z-12=\frac{\sqrt{123}i}{41} z-12=-\frac{\sqrt{123}i}{41}$$

Add 12 to both sides of the equation.

$$z=\frac{\sqrt{123}i}{41}+12 z=-\frac{\sqrt{123}i}{41}+12$$

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 41

$$x ^ 2 -24x +\frac{5907}{41} = 0$$

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

$$r + s = 24 rs = \frac{5907}{41}$$

Two numbers r and s sum up to 24 exactly when the average of the two numbers is \frac{1}{2}*24 = 12. 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>

$$r = 12 - u s = 12 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5907}{41}

$$(12 - u) (12 + u) = \frac{5907}{41}$$

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

$$144 - u^2 = \frac{5907}{41}$$

Simplify the expression by subtracting 144 on both sides

$$-u^2 = \frac{5907}{41}-144 = \frac{3}{41}$$

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

$$u^2 = -\frac{3}{41} u = \pm\sqrt{-\frac{3}{41}} = \pm \frac{\sqrt{3}}{\sqrt{41}}i$$

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

$$r =12 - \frac{\sqrt{3}}{\sqrt{41}}i = 12 - 0.271i s = 12 + \frac{\sqrt{3}}{\sqrt{41}}i = 12 + 0.271i$$