Solve 2x^2+24x-48

$$2 x ^ { 2 } + 24 x - 48$$

Factor

$$2\left(x-\left(-2\sqrt{15}-6\right)\right)\left(x-\left(2\sqrt{15}-6\right)\right)$$

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.

$$2x^{2}+24x-48=0$$

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.

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

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 2\left(-48\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-24±\sqrt{576-8\left(-48\right)}}{2\times 2}$$

Multiply -8 times -48.

$$x=\frac{-24±\sqrt{576+384}}{2\times 2}$$

Add 576 to 384.

$$x=\frac{-24±\sqrt{960}}{2\times 2}$$

Take the square root of 960.

$$x=\frac{-24±8\sqrt{15}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-24±8\sqrt{15}}{4}$$

Now solve the equation x=\frac{-24±8\sqrt{15}}{4} when ± is plus. Add -24 to 8\sqrt{15}.

$$x=\frac{8\sqrt{15}-24}{4}$$

Divide -24+8\sqrt{15} by 4.

$$x=2\sqrt{15}-6$$

Now solve the equation x=\frac{-24±8\sqrt{15}}{4} when ± is minus. Subtract 8\sqrt{15} from -24.

$$x=\frac{-8\sqrt{15}-24}{4}$$

Divide -24-8\sqrt{15} by 4.

$$x=-2\sqrt{15}-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6+2\sqrt{15} for x_{1} and -6-2\sqrt{15} for x_{2}.

$$2x^{2}+24x-48=2\left(x-\left(2\sqrt{15}-6\right)\right)\left(x-\left(-2\sqrt{15}-6\right)\right)$$

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.This is achieved by dividing both sides of the equation by 2

$$x ^ 2 +12x -24 = 0$$

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

$$r + s = -12 rs = -24$$

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 = -24

$$(-6 - u) (-6 + u) = -24$$

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

$$36 - u^2 = -24$$

Simplify the expression by subtracting 36 on both sides

$$-u^2 = -24-36 = -60$$

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

$$u^2 = 60 u = \pm\sqrt{60} = \pm \sqrt{60}$$

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

$$r =-6 - \sqrt{60} = -13.746 s = -6 + \sqrt{60} = 1.746$$

Evaluate

$$2\left(x^{2}+12x-24\right)$$