Solve 3x^2+12x-16

$$3 x ^ { 2 } + 12 x - 16$$

Factor

$$3\left(x-\left(-\frac{2\sqrt{21}}{3}-2\right)\right)\left(x-\left(\frac{2\sqrt{21}}{3}-2\right)\right)$$

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.

$$3x^{2}+12x-16=0$$

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.

$$x=\frac{-12±\sqrt{12^{2}-4\times 3\left(-16\right)}}{2\times 3}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 3\left(-16\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-12±\sqrt{144-12\left(-16\right)}}{2\times 3}$$

Multiply -12 times -16.

$$x=\frac{-12±\sqrt{144+192}}{2\times 3}$$

Add 144 to 192.

$$x=\frac{-12±\sqrt{336}}{2\times 3}$$

Take the square root of 336.

$$x=\frac{-12±4\sqrt{21}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-12±4\sqrt{21}}{6}$$

Now solve the equation x=\frac{-12±4\sqrt{21}}{6} when ± is plus. Add -12 to 4\sqrt{21}.

$$x=\frac{4\sqrt{21}-12}{6}$$

Divide -12+4\sqrt{21} by 6.

$$x=\frac{2\sqrt{21}}{3}-2$$

Now solve the equation x=\frac{-12±4\sqrt{21}}{6} when ± is minus. Subtract 4\sqrt{21} from -12.

$$x=\frac{-4\sqrt{21}-12}{6}$$

Divide -12-4\sqrt{21} by 6.

$$x=-\frac{2\sqrt{21}}{3}-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2+\frac{2\sqrt{21}}{3} for x_{1} and -2-\frac{2\sqrt{21}}{3} for x_{2}.

$$3x^{2}+12x-16=3\left(x-\left(\frac{2\sqrt{21}}{3}-2\right)\right)\left(x-\left(-\frac{2\sqrt{21}}{3}-2\right)\right)$$

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 3

$$x ^ 2 +4x -\frac{16}{3} = 0$$

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

$$r + s = -4 rs = -\frac{16}{3}$$

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{16}{3}

$$(-2 - u) (-2 + u) = -\frac{16}{3}$$

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

$$4 - u^2 = -\frac{16}{3}$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -\frac{16}{3}-4 = -\frac{28}{3}$$

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

$$u^2 = \frac{28}{3} u = \pm\sqrt{\frac{28}{3}} = \pm \frac{\sqrt{28}}{\sqrt{3}}$$

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

$$r =-2 - \frac{\sqrt{28}}{\sqrt{3}} = -5.055 s = -2 + \frac{\sqrt{28}}{\sqrt{3}} = 1.055$$

Evaluate

$$3x^{2}+12x-16$$