Solve 19x^2-230x+620

$$19 x ^ { 2 } - 230 x + 620$$

Factor

$$19\left(x-\frac{115-17\sqrt{5}}{19}\right)\left(x-\frac{17\sqrt{5}+115}{19}\right)$$

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.

$$19x^{2}-230x+620=0$$

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.

$$x=\frac{-\left(-230\right)±\sqrt{\left(-230\right)^{2}-4\times 19\times 620}}{2\times 19}$$

Square -230.

$$x=\frac{-\left(-230\right)±\sqrt{52900-4\times 19\times 620}}{2\times 19}$$

Multiply -4 times 19.

$$x=\frac{-\left(-230\right)±\sqrt{52900-76\times 620}}{2\times 19}$$

Multiply -76 times 620.

$$x=\frac{-\left(-230\right)±\sqrt{52900-47120}}{2\times 19}$$

Add 52900 to -47120.

$$x=\frac{-\left(-230\right)±\sqrt{5780}}{2\times 19}$$

Take the square root of 5780.

$$x=\frac{-\left(-230\right)±34\sqrt{5}}{2\times 19}$$

The opposite of -230 is 230.

$$x=\frac{230±34\sqrt{5}}{2\times 19}$$

Multiply 2 times 19.

$$x=\frac{230±34\sqrt{5}}{38}$$

Now solve the equation x=\frac{230±34\sqrt{5}}{38} when ± is plus. Add 230 to 34\sqrt{5}.

$$x=\frac{34\sqrt{5}+230}{38}$$

Divide 230+34\sqrt{5} by 38.

$$x=\frac{17\sqrt{5}+115}{19}$$

Now solve the equation x=\frac{230±34\sqrt{5}}{38} when ± is minus. Subtract 34\sqrt{5} from 230.

$$x=\frac{230-34\sqrt{5}}{38}$$

Divide 230-34\sqrt{5} by 38.

$$x=\frac{115-17\sqrt{5}}{19}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{115+17\sqrt{5}}{19} for x_{1} and \frac{115-17\sqrt{5}}{19} for x_{2}.

$$19x^{2}-230x+620=19\left(x-\frac{17\sqrt{5}+115}{19}\right)\left(x-\frac{115-17\sqrt{5}}{19}\right)$$

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 19

$$x ^ 2 -\frac{230}{19}x +\frac{620}{19} = 0$$

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

$$r + s = \frac{230}{19} rs = \frac{620}{19}$$

Two numbers r and s sum up to \frac{230}{19} exactly when the average of the two numbers is \frac{1}{2}*\frac{230}{19} = \frac{115}{19}. 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>

$$r = \frac{115}{19} - u s = \frac{115}{19} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{620}{19}

$$(\frac{115}{19} - u) (\frac{115}{19} + u) = \frac{620}{19}$$

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

$$\frac{13225}{361} - u^2 = \frac{620}{19}$$

Simplify the expression by subtracting \frac{13225}{361} on both sides

$$-u^2 = \frac{620}{19}-\frac{13225}{361} = -\frac{1445}{361}$$

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

$$u^2 = \frac{1445}{361} u = \pm\sqrt{\frac{1445}{361}} = \pm \frac{\sqrt{1445}}{19}$$

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

$$r =\frac{115}{19} - \frac{\sqrt{1445}}{19} = 4.052 s = \frac{115}{19} + \frac{\sqrt{1445}}{19} = 8.053$$

Evaluate

$$19x^{2}-230x+620$$