Solve 9x^2-32x-5

$$9 x ^ { 2 } - 32 x - 5$$

Factor

$$9\left(x-\frac{16-\sqrt{301}}{9}\right)\left(x-\frac{\sqrt{301}+16}{9}\right)$$

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.

$$9x^{2}-32x-5=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 9\left(-5\right)}}{2\times 9}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 9\left(-5\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-32\right)±\sqrt{1024-36\left(-5\right)}}{2\times 9}$$

Multiply -36 times -5.

$$x=\frac{-\left(-32\right)±\sqrt{1024+180}}{2\times 9}$$

Add 1024 to 180.

$$x=\frac{-\left(-32\right)±\sqrt{1204}}{2\times 9}$$

Take the square root of 1204.

$$x=\frac{-\left(-32\right)±2\sqrt{301}}{2\times 9}$$

The opposite of -32 is 32.

$$x=\frac{32±2\sqrt{301}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{32±2\sqrt{301}}{18}$$

Now solve the equation x=\frac{32±2\sqrt{301}}{18} when ± is plus. Add 32 to 2\sqrt{301}.

$$x=\frac{2\sqrt{301}+32}{18}$$

Divide 32+2\sqrt{301} by 18.

$$x=\frac{\sqrt{301}+16}{9}$$

Now solve the equation x=\frac{32±2\sqrt{301}}{18} when ± is minus. Subtract 2\sqrt{301} from 32.

$$x=\frac{32-2\sqrt{301}}{18}$$

Divide 32-2\sqrt{301} by 18.

$$x=\frac{16-\sqrt{301}}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{16+\sqrt{301}}{9} for x_{1} and \frac{16-\sqrt{301}}{9} for x_{2}.

$$9x^{2}-32x-5=9\left(x-\frac{\sqrt{301}+16}{9}\right)\left(x-\frac{16-\sqrt{301}}{9}\right)$$

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 9

$$x ^ 2 -\frac{32}{9}x -\frac{5}{9} = 0$$

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

$$r + s = \frac{32}{9} rs = -\frac{5}{9}$$

Two numbers r and s sum up to \frac{32}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{32}{9} = \frac{16}{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>

$$r = \frac{16}{9} - u s = \frac{16}{9} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{9}

$$(\frac{16}{9} - u) (\frac{16}{9} + u) = -\frac{5}{9}$$

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

$$\frac{256}{81} - u^2 = -\frac{5}{9}$$

Simplify the expression by subtracting \frac{256}{81} on both sides

$$-u^2 = -\frac{5}{9}-\frac{256}{81} = -\frac{301}{81}$$

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

$$u^2 = \frac{301}{81} u = \pm\sqrt{\frac{301}{81}} = \pm \frac{\sqrt{301}}{9}$$

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

$$r =\frac{16}{9} - \frac{\sqrt{301}}{9} = -0.150 s = \frac{16}{9} + \frac{\sqrt{301}}{9} = 3.705$$

Evaluate

$$9x^{2}-32x-5$$