Solve -1k^2-16k+12

$$- 1 k ^ { 2 } - 16 k + 12$$

Factor

$$-\left(k-\left(-2\sqrt{19}-8\right)\right)\left(k-\left(2\sqrt{19}-8\right)\right)$$

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.

$$-k^{2}-16k+12=0$$

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.

$$k=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Square -16.

$$k=\frac{-\left(-16\right)±\sqrt{256-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$k=\frac{-\left(-16\right)±\sqrt{256+4\times 12}}{2\left(-1\right)}$$

Multiply 4 times 12.

$$k=\frac{-\left(-16\right)±\sqrt{256+48}}{2\left(-1\right)}$$

Add 256 to 48.

$$k=\frac{-\left(-16\right)±\sqrt{304}}{2\left(-1\right)}$$

Take the square root of 304.

$$k=\frac{-\left(-16\right)±4\sqrt{19}}{2\left(-1\right)}$$

The opposite of -16 is 16.

$$k=\frac{16±4\sqrt{19}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$k=\frac{16±4\sqrt{19}}{-2}$$

Now solve the equation k=\frac{16±4\sqrt{19}}{-2} when ± is plus. Add 16 to 4\sqrt{19}.

$$k=\frac{4\sqrt{19}+16}{-2}$$

Divide 16+4\sqrt{19} by -2.

$$k=-2\sqrt{19}-8$$

Now solve the equation k=\frac{16±4\sqrt{19}}{-2} when ± is minus. Subtract 4\sqrt{19} from 16.

$$k=\frac{16-4\sqrt{19}}{-2}$$

Divide 16-4\sqrt{19} by -2.

$$k=2\sqrt{19}-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -8-2\sqrt{19} for x_{1} and -8+2\sqrt{19} for x_{2}.

$$-k^{2}-16k+12=-\left(k-\left(-2\sqrt{19}-8\right)\right)\left(k-\left(2\sqrt{19}-8\right)\right)$$

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.

$$x ^ 2 +16x -12 = 0$$

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

$$r + s = -16 rs = -12$$

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 = -12

$$(-8 - u) (-8 + u) = -12$$

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

$$64 - u^2 = -12$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = -12-64 = -76$$

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

$$u^2 = 76 u = \pm\sqrt{76} = \pm \sqrt{76}$$

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

$$r =-8 - \sqrt{76} = -16.718 s = -8 + \sqrt{76} = 0.718$$

Evaluate

$$12-16k-k^{2}$$