Solve k^2-8k-5

$$k ^ { 2 } - 8 k - 5$$

Factor

$$\left(k-\left(4-\sqrt{21}\right)\right)\left(k-\left(\sqrt{21}+4\right)\right)$$

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.

$$k^{2}-8k-5=0$$

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.

$$k=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-5\right)}}{2}$$

Square -8.

$$k=\frac{-\left(-8\right)±\sqrt{64-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$k=\frac{-\left(-8\right)±\sqrt{64+20}}{2}$$

Add 64 to 20.

$$k=\frac{-\left(-8\right)±\sqrt{84}}{2}$$

Take the square root of 84.

$$k=\frac{-\left(-8\right)±2\sqrt{21}}{2}$$

The opposite of -8 is 8.

$$k=\frac{8±2\sqrt{21}}{2}$$

Now solve the equation k=\frac{8±2\sqrt{21}}{2} when ± is plus. Add 8 to 2\sqrt{21}.

$$k=\frac{2\sqrt{21}+8}{2}$$

Divide 8+2\sqrt{21} by 2.

$$k=\sqrt{21}+4$$

Now solve the equation k=\frac{8±2\sqrt{21}}{2} when ± is minus. Subtract 2\sqrt{21} from 8.

$$k=\frac{8-2\sqrt{21}}{2}$$

Divide 8-2\sqrt{21} by 2.

$$k=4-\sqrt{21}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4+\sqrt{21} for x_{1} and 4-\sqrt{21} for x_{2}.

$$k^{2}-8k-5=\left(k-\left(\sqrt{21}+4\right)\right)\left(k-\left(4-\sqrt{21}\right)\right)$$

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.

$$x ^ 2 -8x -5 = 0$$

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

$$r + s = 8 rs = -5$$

Two numbers r and s sum up to 8 exactly when the average of the two numbers is \frac{1}{2}*8 = 4. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 4 - u s = 4 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -5

$$(4 - u) (4 + u) = -5$$

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

$$16 - u^2 = -5$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = -5-16 = -21$$

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

$$u^2 = 21 u = \pm\sqrt{21} = \pm \sqrt{21}$$

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

$$r =4 - \sqrt{21} = -0.583 s = 4 + \sqrt{21} = 8.583$$

Evaluate

$$k^{2}-8k-5$$