Solve 2x^2-18x+144

$$2 x ^ { 2 } - 18 x + 144$$

Factor

$$2\left(x^{2}-9x+72\right)$$

Solution Steps

Factor out 2. Polynomial x^{2}-9x+72 is not factored since it does not have any rational roots.

$$2\left(x^{2}-9x+72\right)$$

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.

$$2x^{2}-18x+144=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 2\times 144}}{2\times 2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 2\times 144}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-18\right)±\sqrt{324-8\times 144}}{2\times 2}$$

Multiply -8 times 144.

$$x=\frac{-\left(-18\right)±\sqrt{324-1152}}{2\times 2}$$

Add 324 to -1152.

$$x=\frac{-\left(-18\right)±\sqrt{-828}}{2\times 2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

$$2x^{2}-18x+144$$

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

$$x ^ 2 -9x +72 = 0$$

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

$$r + s = 9 rs = 72$$

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 = 72

$$(\frac{9}{2} - u) (\frac{9}{2} + u) = 72$$

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

$$\frac{81}{4} - u^2 = 72$$

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

$$-u^2 = 72-\frac{81}{4} = \frac{207}{4}$$

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

$$u^2 = -\frac{207}{4} u = \pm\sqrt{-\frac{207}{4}} = \pm \frac{\sqrt{207}}{2}i$$

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

$$r =\frac{9}{2} - \frac{\sqrt{207}}{2}i = 4.500 - 7.194i s = \frac{9}{2} + \frac{\sqrt{207}}{2}i = 4.500 + 7.194i$$

Evaluate

$$2\left(x^{2}-9x+72\right)$$