Solve f(x)=-9x^2-8x+3

$$f ( x ) = - 9 x ^ { 2 } - 8 x + 3$$

Factor

$$-9\left(x-\frac{-\sqrt{43}-4}{9}\right)\left(x-\frac{\sqrt{43}-4}{9}\right)$$

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.

$$-9x^{2}-8x+3=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-9\right)\times 3}}{2\left(-9\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-9\right)\times 3}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-8\right)±\sqrt{64+36\times 3}}{2\left(-9\right)}$$

Multiply 36 times 3.

$$x=\frac{-\left(-8\right)±\sqrt{64+108}}{2\left(-9\right)}$$

Add 64 to 108.

$$x=\frac{-\left(-8\right)±\sqrt{172}}{2\left(-9\right)}$$

Take the square root of 172.

$$x=\frac{-\left(-8\right)±2\sqrt{43}}{2\left(-9\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{43}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{8±2\sqrt{43}}{-18}$$

Now solve the equation x=\frac{8±2\sqrt{43}}{-18} when ± is plus. Add 8 to 2\sqrt{43}.

$$x=\frac{2\sqrt{43}+8}{-18}$$

Divide 8+2\sqrt{43} by -18.

$$x=\frac{-\sqrt{43}-4}{9}$$

Now solve the equation x=\frac{8±2\sqrt{43}}{-18} when ± is minus. Subtract 2\sqrt{43} from 8.

$$x=\frac{8-2\sqrt{43}}{-18}$$

Divide 8-2\sqrt{43} by -18.

$$x=\frac{\sqrt{43}-4}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-4-\sqrt{43}}{9} for x_{1} and \frac{-4+\sqrt{43}}{9} for x_{2}.

$$-9x^{2}-8x+3=-9\left(x-\frac{-\sqrt{43}-4}{9}\right)\left(x-\frac{\sqrt{43}-4}{9}\right)$$

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.

$$x ^ 2 +\frac{8}{9}x -\frac{1}{3} = 0$$

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

$$r + s = -\frac{8}{9} rs = -\frac{1}{3}$$

Two numbers r and s sum up to -\frac{8}{9} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{9} = -\frac{4}{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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{3}

$$(-\frac{4}{9} - u) (-\frac{4}{9} + u) = -\frac{1}{3}$$

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

$$\frac{16}{81} - u^2 = -\frac{1}{3}$$

Simplify the expression by subtracting \frac{16}{81} on both sides

$$-u^2 = -\frac{1}{3}-\frac{16}{81} = -\frac{43}{81}$$

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

$$u^2 = \frac{43}{81} u = \pm\sqrt{\frac{43}{81}} = \pm \frac{\sqrt{43}}{9}$$

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

$$r =-\frac{4}{9} - \frac{\sqrt{43}}{9} = -1.173 s = -\frac{4}{9} + \frac{\sqrt{43}}{9} = 0.284$$

Evaluate

$$3-8x-9x^{2}$$