Solve k^2-24k-48

$$k ^ { 2 } - 24 k - 48$$

Factor

$$\left(k-\left(12-8\sqrt{3}\right)\right)\left(k-\left(8\sqrt{3}+12\right)\right)$$

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.

$$k^{2}-24k-48=0$$

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.

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

Square -24.

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

Multiply -4 times -48.

$$k=\frac{-\left(-24\right)±\sqrt{576+192}}{2}$$

Add 576 to 192.

$$k=\frac{-\left(-24\right)±\sqrt{768}}{2}$$

Take the square root of 768.

$$k=\frac{-\left(-24\right)±16\sqrt{3}}{2}$$

The opposite of -24 is 24.

$$k=\frac{24±16\sqrt{3}}{2}$$

Now solve the equation k=\frac{24±16\sqrt{3}}{2} when ± is plus. Add 24 to 16\sqrt{3}.

$$k=\frac{16\sqrt{3}+24}{2}$$

Divide 24+16\sqrt{3} by 2.

$$k=8\sqrt{3}+12$$

Now solve the equation k=\frac{24±16\sqrt{3}}{2} when ± is minus. Subtract 16\sqrt{3} from 24.

$$k=\frac{24-16\sqrt{3}}{2}$$

Divide 24-16\sqrt{3} by 2.

$$k=12-8\sqrt{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12+8\sqrt{3} for x_{1} and 12-8\sqrt{3} for x_{2}.

$$k^{2}-24k-48=\left(k-\left(8\sqrt{3}+12\right)\right)\left(k-\left(12-8\sqrt{3}\right)\right)$$

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.

$$x ^ 2 -24x -48 = 0$$

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

$$r + s = 24 rs = -48$$

Two numbers r and s sum up to 24 exactly when the average of the two numbers is \frac{1}{2}*24 = 12. 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>

$$r = 12 - u s = 12 + u$$

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

$$(12 - u) (12 + u) = -48$$

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

$$144 - u^2 = -48$$

Simplify the expression by subtracting 144 on both sides

$$-u^2 = -48-144 = -192$$

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

$$u^2 = 192 u = \pm\sqrt{192} = \pm \sqrt{192}$$

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

$$r =12 - \sqrt{192} = -1.856 s = 12 + \sqrt{192} = 25.856$$

Evaluate

$$k^{2}-24k-48$$