Solve 19x^2+60x-173

$$19 x ^ { 2 } + 60 x - 173$$

Factor

$$19\left(x-\frac{-\sqrt{4187}-30}{19}\right)\left(x-\frac{\sqrt{4187}-30}{19}\right)$$

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.

$$19x^{2}+60x-173=0$$

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.

$$x=\frac{-60±\sqrt{60^{2}-4\times 19\left(-173\right)}}{2\times 19}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\times 19\left(-173\right)}}{2\times 19}$$

Multiply -4 times 19.

$$x=\frac{-60±\sqrt{3600-76\left(-173\right)}}{2\times 19}$$

Multiply -76 times -173.

$$x=\frac{-60±\sqrt{3600+13148}}{2\times 19}$$

Add 3600 to 13148.

$$x=\frac{-60±\sqrt{16748}}{2\times 19}$$

Take the square root of 16748.

$$x=\frac{-60±2\sqrt{4187}}{2\times 19}$$

Multiply 2 times 19.

$$x=\frac{-60±2\sqrt{4187}}{38}$$

Now solve the equation x=\frac{-60±2\sqrt{4187}}{38} when ± is plus. Add -60 to 2\sqrt{4187}.

$$x=\frac{2\sqrt{4187}-60}{38}$$

Divide -60+2\sqrt{4187} by 38.

$$x=\frac{\sqrt{4187}-30}{19}$$

Now solve the equation x=\frac{-60±2\sqrt{4187}}{38} when ± is minus. Subtract 2\sqrt{4187} from -60.

$$x=\frac{-2\sqrt{4187}-60}{38}$$

Divide -60-2\sqrt{4187} by 38.

$$x=\frac{-\sqrt{4187}-30}{19}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-30+\sqrt{4187}}{19} for x_{1} and \frac{-30-\sqrt{4187}}{19} for x_{2}.

$$19x^{2}+60x-173=19\left(x-\frac{\sqrt{4187}-30}{19}\right)\left(x-\frac{-\sqrt{4187}-30}{19}\right)$$

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

$$x ^ 2 +\frac{60}{19}x -\frac{173}{19} = 0$$

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

$$r + s = -\frac{60}{19} rs = -\frac{173}{19}$$

Two numbers r and s sum up to -\frac{60}{19} exactly when the average of the two numbers is \frac{1}{2}*-\frac{60}{19} = -\frac{30}{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>

$$r = -\frac{30}{19} - u s = -\frac{30}{19} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{173}{19}

$$(-\frac{30}{19} - u) (-\frac{30}{19} + u) = -\frac{173}{19}$$

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

$$\frac{900}{361} - u^2 = -\frac{173}{19}$$

Simplify the expression by subtracting \frac{900}{361} on both sides

$$-u^2 = -\frac{173}{19}-\frac{900}{361} = -\frac{4187}{361}$$

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

$$u^2 = \frac{4187}{361} u = \pm\sqrt{\frac{4187}{361}} = \pm \frac{\sqrt{4187}}{19}$$

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

$$r =-\frac{30}{19} - \frac{\sqrt{4187}}{19} = -4.985 s = -\frac{30}{19} + \frac{\sqrt{4187}}{19} = 1.827$$

Evaluate

$$19x^{2}+60x-173$$