Solve x^2-16x-48

$$x ^ { 2 } - 16 x - 48$$

Factor

$$\left(x-\left(8-4\sqrt{7}\right)\right)\left(x-\left(4\sqrt{7}+8\right)\right)$$

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.

$$x^{2}-16x-48=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-48\right)}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-48\right)}}{2}$$

Multiply -4 times -48.

$$x=\frac{-\left(-16\right)±\sqrt{256+192}}{2}$$

Add 256 to 192.

$$x=\frac{-\left(-16\right)±\sqrt{448}}{2}$$

Take the square root of 448.

$$x=\frac{-\left(-16\right)±8\sqrt{7}}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±8\sqrt{7}}{2}$$

Now solve the equation x=\frac{16±8\sqrt{7}}{2} when ± is plus. Add 16 to 8\sqrt{7}.

$$x=\frac{8\sqrt{7}+16}{2}$$

Divide 16+8\sqrt{7} by 2.

$$x=4\sqrt{7}+8$$

Now solve the equation x=\frac{16±8\sqrt{7}}{2} when ± is minus. Subtract 8\sqrt{7} from 16.

$$x=\frac{16-8\sqrt{7}}{2}$$

Divide 16-8\sqrt{7} by 2.

$$x=8-4\sqrt{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8+4\sqrt{7} for x_{1} and 8-4\sqrt{7} for x_{2}.

$$x^{2}-16x-48=\left(x-\left(4\sqrt{7}+8\right)\right)\left(x-\left(8-4\sqrt{7}\right)\right)$$

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.

$$x ^ 2 -16x -48 = 0$$

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

$$r + s = 16 rs = -48$$

Two numbers r and s sum up to 16 exactly when the average of the two numbers is \frac{1}{2}*16 = 8. 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>

$$r = 8 - u s = 8 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -48

$$(8 - u) (8 + u) = -48$$

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

$$64 - u^2 = -48$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = -48-64 = -112$$

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

$$u^2 = 112 u = \pm\sqrt{112} = \pm \sqrt{112}$$

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

$$r =8 - \sqrt{112} = -2.583 s = 8 + \sqrt{112} = 18.583$$

Evaluate

$$x^{2}-16x-48$$