Solve f(x)=-9x^2+4x+9

$$f ( x ) = - 9 x ^ { 2 } + 4 x + 9$$

Factor

$$-9\left(x-\frac{2-\sqrt{85}}{9}\right)\left(x-\frac{\sqrt{85}+2}{9}\right)$$

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.

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

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.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-9\right)\times 9}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-4±\sqrt{16+36\times 9}}{2\left(-9\right)}$$

Multiply 36 times 9.

$$x=\frac{-4±\sqrt{16+324}}{2\left(-9\right)}$$

Add 16 to 324.

$$x=\frac{-4±\sqrt{340}}{2\left(-9\right)}$$

Take the square root of 340.

$$x=\frac{-4±2\sqrt{85}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-4±2\sqrt{85}}{-18}$$

Now solve the equation x=\frac{-4±2\sqrt{85}}{-18} when ± is plus. Add -4 to 2\sqrt{85}.

$$x=\frac{2\sqrt{85}-4}{-18}$$

Divide -4+2\sqrt{85} by -18.

$$x=\frac{2-\sqrt{85}}{9}$$

Now solve the equation x=\frac{-4±2\sqrt{85}}{-18} when ± is minus. Subtract 2\sqrt{85} from -4.

$$x=\frac{-2\sqrt{85}-4}{-18}$$

Divide -4-2\sqrt{85} by -18.

$$x=\frac{\sqrt{85}+2}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2-\sqrt{85}}{9} for x_{1} and \frac{2+\sqrt{85}}{9} for x_{2}.

$$-9x^{2}+4x+9=-9\left(x-\frac{2-\sqrt{85}}{9}\right)\left(x-\frac{\sqrt{85}+2}{9}\right)$$

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.

$$x ^ 2 -\frac{4}{9}x -1 = 0$$

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

$$r + s = \frac{4}{9} rs = -1$$

Two numbers r and s sum up to \frac{4}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{4}{9} = \frac{2}{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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(\frac{2}{9} - u) (\frac{2}{9} + u) = -1$$

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

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

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

$$-u^2 = -1-\frac{4}{81} = -\frac{85}{81}$$

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

$$u^2 = \frac{85}{81} u = \pm\sqrt{\frac{85}{81}} = \pm \frac{\sqrt{85}}{9}$$

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

$$r =\frac{2}{9} - \frac{\sqrt{85}}{9} = -0.802 s = \frac{2}{9} + \frac{\sqrt{85}}{9} = 1.247$$

Evaluate

$$9+4x-9x^{2}$$