Solve 36x^2+8x-16=

$$36 x ^ { 2 } + 8 x - 16 =$$

Factor

$$36\left(x-\frac{-\sqrt{37}-1}{9}\right)\left(x-\frac{\sqrt{37}-1}{9}\right)$$

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.

$$36x^{2}+8x-16=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\times 36\left(-16\right)}}{2\times 36}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 36\left(-16\right)}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-8±\sqrt{64-144\left(-16\right)}}{2\times 36}$$

Multiply -144 times -16.

$$x=\frac{-8±\sqrt{64+2304}}{2\times 36}$$

Add 64 to 2304.

$$x=\frac{-8±\sqrt{2368}}{2\times 36}$$

Take the square root of 2368.

$$x=\frac{-8±8\sqrt{37}}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{-8±8\sqrt{37}}{72}$$

Now solve the equation x=\frac{-8±8\sqrt{37}}{72} when ± is plus. Add -8 to 8\sqrt{37}.

$$x=\frac{8\sqrt{37}-8}{72}$$

Divide -8+8\sqrt{37} by 72.

$$x=\frac{\sqrt{37}-1}{9}$$

Now solve the equation x=\frac{-8±8\sqrt{37}}{72} when ± is minus. Subtract 8\sqrt{37} from -8.

$$x=\frac{-8\sqrt{37}-8}{72}$$

Divide -8-8\sqrt{37} by 72.

$$x=\frac{-\sqrt{37}-1}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{37}}{9} for x_{1} and \frac{-1-\sqrt{37}}{9} for x_{2}.

$$36x^{2}+8x-16=36\left(x-\frac{\sqrt{37}-1}{9}\right)\left(x-\frac{-\sqrt{37}-1}{9}\right)$$

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 36

$$x ^ 2 +\frac{2}{9}x -\frac{4}{9} = 0$$

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

$$r + s = -\frac{2}{9} rs = -\frac{4}{9}$$

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

$$r = -\frac{1}{9} - u s = -\frac{1}{9} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{9}

$$(-\frac{1}{9} - u) (-\frac{1}{9} + u) = -\frac{4}{9}$$

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

$$\frac{1}{81} - u^2 = -\frac{4}{9}$$

Simplify the expression by subtracting \frac{1}{81} on both sides

$$-u^2 = -\frac{4}{9}-\frac{1}{81} = -\frac{37}{81}$$

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

$$u^2 = \frac{37}{81} u = \pm\sqrt{\frac{37}{81}} = \pm \frac{\sqrt{37}}{9}$$

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

$$r =-\frac{1}{9} - \frac{\sqrt{37}}{9} = -0.787 s = -\frac{1}{9} + \frac{\sqrt{37}}{9} = 0.565$$

Evaluate

$$36x^{2}+8x-16$$