Solve 19x^2-88x+33

$$19 x ^ { 2 } - 88 x + 33$$

Factor

$$19\left(x-\frac{44-\sqrt{1309}}{19}\right)\left(x-\frac{\sqrt{1309}+44}{19}\right)$$

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.

$$19x^{2}-88x+33=0$$

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.

$$x=\frac{-\left(-88\right)±\sqrt{\left(-88\right)^{2}-4\times 19\times 33}}{2\times 19}$$

Square -88.

$$x=\frac{-\left(-88\right)±\sqrt{7744-4\times 19\times 33}}{2\times 19}$$

Multiply -4 times 19.

$$x=\frac{-\left(-88\right)±\sqrt{7744-76\times 33}}{2\times 19}$$

Multiply -76 times 33.

$$x=\frac{-\left(-88\right)±\sqrt{7744-2508}}{2\times 19}$$

Add 7744 to -2508.

$$x=\frac{-\left(-88\right)±\sqrt{5236}}{2\times 19}$$

Take the square root of 5236.

$$x=\frac{-\left(-88\right)±2\sqrt{1309}}{2\times 19}$$

The opposite of -88 is 88.

$$x=\frac{88±2\sqrt{1309}}{2\times 19}$$

Multiply 2 times 19.

$$x=\frac{88±2\sqrt{1309}}{38}$$

Now solve the equation x=\frac{88±2\sqrt{1309}}{38} when ± is plus. Add 88 to 2\sqrt{1309}.

$$x=\frac{2\sqrt{1309}+88}{38}$$

Divide 88+2\sqrt{1309} by 38.

$$x=\frac{\sqrt{1309}+44}{19}$$

Now solve the equation x=\frac{88±2\sqrt{1309}}{38} when ± is minus. Subtract 2\sqrt{1309} from 88.

$$x=\frac{88-2\sqrt{1309}}{38}$$

Divide 88-2\sqrt{1309} by 38.

$$x=\frac{44-\sqrt{1309}}{19}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{44+\sqrt{1309}}{19} for x_{1} and \frac{44-\sqrt{1309}}{19} for x_{2}.

$$19x^{2}-88x+33=19\left(x-\frac{\sqrt{1309}+44}{19}\right)\left(x-\frac{44-\sqrt{1309}}{19}\right)$$

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

$$x ^ 2 -\frac{88}{19}x +\frac{33}{19} = 0$$

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

$$r + s = \frac{88}{19} rs = \frac{33}{19}$$

Two numbers r and s sum up to \frac{88}{19} exactly when the average of the two numbers is \frac{1}{2}*\frac{88}{19} = \frac{44}{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>

$$r = \frac{44}{19} - u s = \frac{44}{19} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{33}{19}

$$(\frac{44}{19} - u) (\frac{44}{19} + u) = \frac{33}{19}$$

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

$$\frac{1936}{361} - u^2 = \frac{33}{19}$$

Simplify the expression by subtracting \frac{1936}{361} on both sides

$$-u^2 = \frac{33}{19}-\frac{1936}{361} = -\frac{1309}{361}$$

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

$$u^2 = \frac{1309}{361} u = \pm\sqrt{\frac{1309}{361}} = \pm \frac{\sqrt{1309}}{19}$$

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

$$r =\frac{44}{19} - \frac{\sqrt{1309}}{19} = 0.412 s = \frac{44}{19} + \frac{\sqrt{1309}}{19} = 4.220$$

Evaluate

$$19x^{2}-88x+33$$