Solve 9x^2+36x-72

$$9 x ^ { 2 } + 36 x - 72$$

Factor

$$9\left(x-\left(-2\sqrt{3}-2\right)\right)\left(x-\left(2\sqrt{3}-2\right)\right)$$

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.

$$9x^{2}+36x-72=0$$

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.

$$x=\frac{-36±\sqrt{36^{2}-4\times 9\left(-72\right)}}{2\times 9}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\times 9\left(-72\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-36±\sqrt{1296-36\left(-72\right)}}{2\times 9}$$

Multiply -36 times -72.

$$x=\frac{-36±\sqrt{1296+2592}}{2\times 9}$$

Add 1296 to 2592.

$$x=\frac{-36±\sqrt{3888}}{2\times 9}$$

Take the square root of 3888.

$$x=\frac{-36±36\sqrt{3}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-36±36\sqrt{3}}{18}$$

Now solve the equation x=\frac{-36±36\sqrt{3}}{18} when ± is plus. Add -36 to 36\sqrt{3}.

$$x=\frac{36\sqrt{3}-36}{18}$$

Divide -36+36\sqrt{3} by 18.

$$x=2\sqrt{3}-2$$

Now solve the equation x=\frac{-36±36\sqrt{3}}{18} when ± is minus. Subtract 36\sqrt{3} from -36.

$$x=\frac{-36\sqrt{3}-36}{18}$$

Divide -36-36\sqrt{3} by 18.

$$x=-2\sqrt{3}-2$$

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

$$9x^{2}+36x-72=9\left(x-\left(2\sqrt{3}-2\right)\right)\left(x-\left(-2\sqrt{3}-2\right)\right)$$

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.This is achieved by dividing both sides of the equation by 9

$$x ^ 2 +4x -8 = 0$$

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

$$r + s = -4 rs = -8$$

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

$$r = -2 - u s = -2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -8

$$(-2 - u) (-2 + u) = -8$$

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

$$4 - u^2 = -8$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -8-4 = -12$$

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

$$u^2 = 12 u = \pm\sqrt{12} = \pm \sqrt{12}$$

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

$$r =-2 - \sqrt{12} = -5.464 s = -2 + \sqrt{12} = 1.464$$

Evaluate

$$9\left(x^{2}+4x-8\right)$$