Solve {r}{-4x^2-48x}{+36}

$$\left. \begin{array} { r } { - 4 x ^ { 2 } - 48 x } \\ { + 36 } \end{array} \right.$$

Factor

$$-4\left(x-\left(-3\sqrt{5}-6\right)\right)\left(x-\left(3\sqrt{5}-6\right)\right)$$

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.

$$-4x^{2}-48x+36=0$$

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.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\left(-4\right)\times 36}}{2\left(-4\right)}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\left(-4\right)\times 36}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-48\right)±\sqrt{2304+16\times 36}}{2\left(-4\right)}$$

Multiply 16 times 36.

$$x=\frac{-\left(-48\right)±\sqrt{2304+576}}{2\left(-4\right)}$$

Add 2304 to 576.

$$x=\frac{-\left(-48\right)±\sqrt{2880}}{2\left(-4\right)}$$

Take the square root of 2880.

$$x=\frac{-\left(-48\right)±24\sqrt{5}}{2\left(-4\right)}$$

The opposite of -48 is 48.

$$x=\frac{48±24\sqrt{5}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{48±24\sqrt{5}}{-8}$$

Now solve the equation x=\frac{48±24\sqrt{5}}{-8} when ± is plus. Add 48 to 24\sqrt{5}.

$$x=\frac{24\sqrt{5}+48}{-8}$$

Divide 48+24\sqrt{5} by -8.

$$x=-3\sqrt{5}-6$$

Now solve the equation x=\frac{48±24\sqrt{5}}{-8} when ± is minus. Subtract 24\sqrt{5} from 48.

$$x=\frac{48-24\sqrt{5}}{-8}$$

Divide 48-24\sqrt{5} by -8.

$$x=3\sqrt{5}-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6-3\sqrt{5} for x_{1} and -6+3\sqrt{5} for x_{2}.

$$-4x^{2}-48x+36=-4\left(x-\left(-3\sqrt{5}-6\right)\right)\left(x-\left(3\sqrt{5}-6\right)\right)$$

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.

$$x ^ 2 +12x -9 = 0$$

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

$$r + s = -12 rs = -9$$

Two numbers r and s sum up to -12 exactly when the average of the two numbers is \frac{1}{2}*-12 = -6. 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>

$$r = -6 - u s = -6 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -9

$$(-6 - u) (-6 + u) = -9$$

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

$$36 - u^2 = -9$$

Simplify the expression by subtracting 36 on both sides

$$-u^2 = -9-36 = -45$$

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

$$u^2 = 45 u = \pm\sqrt{45} = \pm \sqrt{45}$$

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

$$r =-6 - \sqrt{45} = -12.708 s = -6 + \sqrt{45} = 0.708$$

Evaluate

$$36-48x-4x^{2}$$