Solve x^2-1826x+46800

$$x ^ { 2 } - 1826 x + 46800$$

Factor

$$\left(x-1800\right)\left(x-26\right)$$

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.

$$x^{2}-1826x+46800=0$$

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.

$$x=\frac{-\left(-1826\right)±\sqrt{\left(-1826\right)^{2}-4\times 46800}}{2}$$

Square -1826.

$$x=\frac{-\left(-1826\right)±\sqrt{3334276-4\times 46800}}{2}$$

Multiply -4 times 46800.

$$x=\frac{-\left(-1826\right)±\sqrt{3334276-187200}}{2}$$

Add 3334276 to -187200.

$$x=\frac{-\left(-1826\right)±\sqrt{3147076}}{2}$$

Take the square root of 3147076.

$$x=\frac{-\left(-1826\right)±1774}{2}$$

The opposite of -1826 is 1826.

$$x=\frac{1826±1774}{2}$$

Now solve the equation x=\frac{1826±1774}{2} when ± is plus. Add 1826 to 1774.

$$x=\frac{3600}{2}$$

Divide 3600 by 2.

$$x=1800$$

Now solve the equation x=\frac{1826±1774}{2} when ± is minus. Subtract 1774 from 1826.

$$x=\frac{52}{2}$$

Divide 52 by 2.

$$x=26$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1800 for x_{1} and 26 for x_{2}.

$$x^{2}-1826x+46800=\left(x-1800\right)\left(x-26\right)$$

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.

$$x ^ 2 -1826x +46800 = 0$$

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

$$r + s = 1826 rs = 46800$$

Two numbers r and s sum up to 1826 exactly when the average of the two numbers is \frac{1}{2}*1826 = 913. 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>

$$r = 913 - u s = 913 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 46800

$$(913 - u) (913 + u) = 46800$$

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

$$833569 - u^2 = 46800$$

Simplify the expression by subtracting 833569 on both sides

$$-u^2 = 46800-833569 = -786769$$

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

$$u^2 = 786769 u = \pm\sqrt{786769} = \pm 887$$

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

$$r =913 - 887 = 26 s = 913 + 887 = 1800$$

Evaluate

$$\left(x-1800\right)\left(x-26\right)$$