Solve 11x^2-1230x+5600

$$11 x ^ { 2 } - 1230 x + 5600$$

Factor

$$11\left(x-\frac{615-5\sqrt{12665}}{11}\right)\left(x-\frac{5\sqrt{12665}+615}{11}\right)$$

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.

$$11x^{2}-1230x+5600=0$$

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.

$$x=\frac{-\left(-1230\right)±\sqrt{\left(-1230\right)^{2}-4\times 11\times 5600}}{2\times 11}$$

Square -1230.

$$x=\frac{-\left(-1230\right)±\sqrt{1512900-4\times 11\times 5600}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-1230\right)±\sqrt{1512900-44\times 5600}}{2\times 11}$$

Multiply -44 times 5600.

$$x=\frac{-\left(-1230\right)±\sqrt{1512900-246400}}{2\times 11}$$

Add 1512900 to -246400.

$$x=\frac{-\left(-1230\right)±\sqrt{1266500}}{2\times 11}$$

Take the square root of 1266500.

$$x=\frac{-\left(-1230\right)±10\sqrt{12665}}{2\times 11}$$

The opposite of -1230 is 1230.

$$x=\frac{1230±10\sqrt{12665}}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{1230±10\sqrt{12665}}{22}$$

Now solve the equation x=\frac{1230±10\sqrt{12665}}{22} when ± is plus. Add 1230 to 10\sqrt{12665}.

$$x=\frac{10\sqrt{12665}+1230}{22}$$

Divide 1230+10\sqrt{12665} by 22.

$$x=\frac{5\sqrt{12665}+615}{11}$$

Now solve the equation x=\frac{1230±10\sqrt{12665}}{22} when ± is minus. Subtract 10\sqrt{12665} from 1230.

$$x=\frac{1230-10\sqrt{12665}}{22}$$

Divide 1230-10\sqrt{12665} by 22.

$$x=\frac{615-5\sqrt{12665}}{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{615+5\sqrt{12665}}{11} for x_{1} and \frac{615-5\sqrt{12665}}{11} for x_{2}.

$$11x^{2}-1230x+5600=11\left(x-\frac{5\sqrt{12665}+615}{11}\right)\left(x-\frac{615-5\sqrt{12665}}{11}\right)$$

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

$$x ^ 2 -\frac{1230}{11}x +\frac{5600}{11} = 0$$

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

$$r + s = \frac{1230}{11} rs = \frac{5600}{11}$$

Two numbers r and s sum up to \frac{1230}{11} exactly when the average of the two numbers is \frac{1}{2}*\frac{1230}{11} = \frac{615}{11}. 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>

$$r = \frac{615}{11} - u s = \frac{615}{11} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5600}{11}

$$(\frac{615}{11} - u) (\frac{615}{11} + u) = \frac{5600}{11}$$

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

$$\frac{378225}{121} - u^2 = \frac{5600}{11}$$

Simplify the expression by subtracting \frac{378225}{121} on both sides

$$-u^2 = \frac{5600}{11}-\frac{378225}{121} = -\frac{316625}{121}$$

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

$$u^2 = \frac{316625}{121} u = \pm\sqrt{\frac{316625}{121}} = \pm \frac{\sqrt{316625}}{11}$$

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

$$r =\frac{615}{11} - \frac{\sqrt{316625}}{11} = 4.755 s = \frac{615}{11} + \frac{\sqrt{316625}}{11} = 107.063$$

Evaluate

$$11x^{2}-1230x+5600$$