Solve 24c^2+96c+9

$$24 c ^ { 2 } + 96 c + 9$$

Factor

$$24\left(c-\left(-\frac{\sqrt{58}}{4}-2\right)\right)\left(c-\left(\frac{\sqrt{58}}{4}-2\right)\right)$$

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.

$$24c^{2}+96c+9=0$$

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.

$$c=\frac{-96±\sqrt{96^{2}-4\times 24\times 9}}{2\times 24}$$

Square 96.

$$c=\frac{-96±\sqrt{9216-4\times 24\times 9}}{2\times 24}$$

Multiply -4 times 24.

$$c=\frac{-96±\sqrt{9216-96\times 9}}{2\times 24}$$

Multiply -96 times 9.

$$c=\frac{-96±\sqrt{9216-864}}{2\times 24}$$

Add 9216 to -864.

$$c=\frac{-96±\sqrt{8352}}{2\times 24}$$

Take the square root of 8352.

$$c=\frac{-96±12\sqrt{58}}{2\times 24}$$

Multiply 2 times 24.

$$c=\frac{-96±12\sqrt{58}}{48}$$

Now solve the equation c=\frac{-96±12\sqrt{58}}{48} when ± is plus. Add -96 to 12\sqrt{58}.

$$c=\frac{12\sqrt{58}-96}{48}$$

Divide -96+12\sqrt{58} by 48.

$$c=\frac{\sqrt{58}}{4}-2$$

Now solve the equation c=\frac{-96±12\sqrt{58}}{48} when ± is minus. Subtract 12\sqrt{58} from -96.

$$c=\frac{-12\sqrt{58}-96}{48}$$

Divide -96-12\sqrt{58} by 48.

$$c=-\frac{\sqrt{58}}{4}-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2+\frac{\sqrt{58}}{4} for x_{1} and -2-\frac{\sqrt{58}}{4} for x_{2}.

$$24c^{2}+96c+9=24\left(c-\left(\frac{\sqrt{58}}{4}-2\right)\right)\left(c-\left(-\frac{\sqrt{58}}{4}-2\right)\right)$$

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 24

$$x ^ 2 +4x +\frac{3}{8} = 0$$

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

$$r + s = -4 rs = \frac{3}{8}$$

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 = \frac{3}{8}

$$(-2 - u) (-2 + u) = \frac{3}{8}$$

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

$$4 - u^2 = \frac{3}{8}$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = \frac{3}{8}-4 = -\frac{29}{8}$$

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

$$u^2 = \frac{29}{8} u = \pm\sqrt{\frac{29}{8}} = \pm \frac{\sqrt{29}}{\sqrt{8}}$$

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

$$r =-2 - \frac{\sqrt{29}}{\sqrt{8}} = -3.904 s = -2 + \frac{\sqrt{29}}{\sqrt{8}} = -0.096$$

Evaluate

$$24c^{2}+96c+9$$