Solve 1x^2-492x+4320

$$1 x ^ { 2 } - 492 x + 4320$$

Factor

$$\left(x-\left(246-6\sqrt{1561}\right)\right)\left(x-\left(6\sqrt{1561}+246\right)\right)$$

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.

$$x^{2}-492x+4320=0$$

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.

$$x=\frac{-\left(-492\right)±\sqrt{\left(-492\right)^{2}-4\times 4320}}{2}$$

Square -492.

$$x=\frac{-\left(-492\right)±\sqrt{242064-4\times 4320}}{2}$$

Multiply -4 times 4320.

$$x=\frac{-\left(-492\right)±\sqrt{242064-17280}}{2}$$

Add 242064 to -17280.

$$x=\frac{-\left(-492\right)±\sqrt{224784}}{2}$$

Take the square root of 224784.

$$x=\frac{-\left(-492\right)±12\sqrt{1561}}{2}$$

The opposite of -492 is 492.

$$x=\frac{492±12\sqrt{1561}}{2}$$

Now solve the equation x=\frac{492±12\sqrt{1561}}{2} when ± is plus. Add 492 to 12\sqrt{1561}.

$$x=\frac{12\sqrt{1561}+492}{2}$$

Divide 492+12\sqrt{1561} by 2.

$$x=6\sqrt{1561}+246$$

Now solve the equation x=\frac{492±12\sqrt{1561}}{2} when ± is minus. Subtract 12\sqrt{1561} from 492.

$$x=\frac{492-12\sqrt{1561}}{2}$$

Divide 492-12\sqrt{1561} by 2.

$$x=246-6\sqrt{1561}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 246+6\sqrt{1561} for x_{1} and 246-6\sqrt{1561} for x_{2}.

$$x^{2}-492x+4320=\left(x-\left(6\sqrt{1561}+246\right)\right)\left(x-\left(246-6\sqrt{1561}\right)\right)$$

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.

$$x ^ 2 -492x +4320 = 0$$

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

$$r + s = 492 rs = 4320$$

Two numbers r and s sum up to 492 exactly when the average of the two numbers is \frac{1}{2}*492 = 246. 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>

$$r = 246 - u s = 246 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 4320

$$(246 - u) (246 + u) = 4320$$

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

$$60516 - u^2 = 4320$$

Simplify the expression by subtracting 60516 on both sides

$$-u^2 = 4320-60516 = -56196$$

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

$$u^2 = 56196 u = \pm\sqrt{56196} = \pm \sqrt{56196}$$

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

$$r =246 - \sqrt{56196} = 8.943 s = 246 + \sqrt{56196} = 483.057$$

Evaluate

$$x^{2}-492x+4320$$