Solve 2x^2-18x+20

$$2 x ^ { 2 } - 18 x + 20$$

Factor

$$2\left(x-\frac{9-\sqrt{41}}{2}\right)\left(x-\frac{\sqrt{41}+9}{2}\right)$$

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.

$$2x^{2}-18x+20=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 2\times 20}}{2\times 2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 2\times 20}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-18\right)±\sqrt{324-8\times 20}}{2\times 2}$$

Multiply -8 times 20.

$$x=\frac{-\left(-18\right)±\sqrt{324-160}}{2\times 2}$$

Add 324 to -160.

$$x=\frac{-\left(-18\right)±\sqrt{164}}{2\times 2}$$

Take the square root of 164.

$$x=\frac{-\left(-18\right)±2\sqrt{41}}{2\times 2}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{41}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{18±2\sqrt{41}}{4}$$

Now solve the equation x=\frac{18±2\sqrt{41}}{4} when ± is plus. Add 18 to 2\sqrt{41}.

$$x=\frac{2\sqrt{41}+18}{4}$$

Divide 18+2\sqrt{41} by 4.

$$x=\frac{\sqrt{41}+9}{2}$$

Now solve the equation x=\frac{18±2\sqrt{41}}{4} when ± is minus. Subtract 2\sqrt{41} from 18.

$$x=\frac{18-2\sqrt{41}}{4}$$

Divide 18-2\sqrt{41} by 4.

$$x=\frac{9-\sqrt{41}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9+\sqrt{41}}{2} for x_{1} and \frac{9-\sqrt{41}}{2} for x_{2}.

$$2x^{2}-18x+20=2\left(x-\frac{\sqrt{41}+9}{2}\right)\left(x-\frac{9-\sqrt{41}}{2}\right)$$

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 2

$$x ^ 2 -9x +10 = 0$$

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

$$r + s = 9 rs = 10$$

Two numbers r and s sum up to 9 exactly when the average of the two numbers is \frac{1}{2}*9 = \frac{9}{2}. 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>

$$r = \frac{9}{2} - u s = \frac{9}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 10

$$(\frac{9}{2} - u) (\frac{9}{2} + u) = 10$$

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

$$\frac{81}{4} - u^2 = 10$$

Simplify the expression by subtracting \frac{81}{4} on both sides

$$-u^2 = 10-\frac{81}{4} = -\frac{41}{4}$$

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

$$u^2 = \frac{41}{4} u = \pm\sqrt{\frac{41}{4}} = \pm \frac{\sqrt{41}}{2}$$

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

$$r =\frac{9}{2} - \frac{\sqrt{41}}{2} = 1.298 s = \frac{9}{2} + \frac{\sqrt{41}}{2} = 7.702$$

Evaluate

$$2\left(x^{2}-9x+10\right)$$