Solve -9x^2+18x+68=

$$- 9 x ^ { 2 } + 18 x + 68 =$$

Factor

$$-9\left(x-\left(-\frac{\sqrt{77}}{3}+1\right)\right)\left(x-\left(\frac{\sqrt{77}}{3}+1\right)\right)$$

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.

$$-9x^{2}+18x+68=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-9\right)\times 68}}{2\left(-9\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-9\right)\times 68}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-18±\sqrt{324+36\times 68}}{2\left(-9\right)}$$

Multiply 36 times 68.

$$x=\frac{-18±\sqrt{324+2448}}{2\left(-9\right)}$$

Add 324 to 2448.

$$x=\frac{-18±\sqrt{2772}}{2\left(-9\right)}$$

Take the square root of 2772.

$$x=\frac{-18±6\sqrt{77}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-18±6\sqrt{77}}{-18}$$

Now solve the equation x=\frac{-18±6\sqrt{77}}{-18} when ± is plus. Add -18 to 6\sqrt{77}.

$$x=\frac{6\sqrt{77}-18}{-18}$$

Divide -18+6\sqrt{77} by -18.

$$x=-\frac{\sqrt{77}}{3}+1$$

Now solve the equation x=\frac{-18±6\sqrt{77}}{-18} when ± is minus. Subtract 6\sqrt{77} from -18.

$$x=\frac{-6\sqrt{77}-18}{-18}$$

Divide -18-6\sqrt{77} by -18.

$$x=\frac{\sqrt{77}}{3}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\frac{\sqrt{77}}{3} for x_{1} and 1+\frac{\sqrt{77}}{3} for x_{2}.

$$-9x^{2}+18x+68=-9\left(x-\left(-\frac{\sqrt{77}}{3}+1\right)\right)\left(x-\left(\frac{\sqrt{77}}{3}+1\right)\right)$$

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.

$$x ^ 2 -2x -\frac{68}{9} = 0$$

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

$$r + s = 2 rs = -\frac{68}{9}$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{68}{9}

$$(1 - u) (1 + u) = -\frac{68}{9}$$

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

$$1 - u^2 = -\frac{68}{9}$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -\frac{68}{9}-1 = -\frac{77}{9}$$

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

$$u^2 = \frac{77}{9} u = \pm\sqrt{\frac{77}{9}} = \pm \frac{\sqrt{77}}{3}$$

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

$$r =1 - \frac{\sqrt{77}}{3} = -1.925 s = 1 + \frac{\sqrt{77}}{3} = 3.925$$

Evaluate

$$68+18x-9x^{2}$$