Solve f(x)=-x^2-8x+1

$$f ( x ) = - x ^ { 2 } - 8 x + 1$$

Factor

$$-\left(x-\left(-\sqrt{17}-4\right)\right)\left(x-\left(\sqrt{17}-4\right)\right)$$

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.

$$-x^{2}-8x+1=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4}}{2\left(-1\right)}$$

Add 64 to 4.

$$x=\frac{-\left(-8\right)±\sqrt{68}}{2\left(-1\right)}$$

Take the square root of 68.

$$x=\frac{-\left(-8\right)±2\sqrt{17}}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{8±2\sqrt{17}}{-2}$$

Now solve the equation x=\frac{8±2\sqrt{17}}{-2} when ± is plus. Add 8 to 2\sqrt{17}.

$$x=\frac{2\sqrt{17}+8}{-2}$$

Divide 8+2\sqrt{17} by -2.

$$x=-\left(\sqrt{17}+4\right)$$

Now solve the equation x=\frac{8±2\sqrt{17}}{-2} when ± is minus. Subtract 2\sqrt{17} from 8.

$$x=\frac{8-2\sqrt{17}}{-2}$$

Divide 8-2\sqrt{17} by -2.

$$x=\sqrt{17}-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\left(4+\sqrt{17}\right) for x_{1} and -4+\sqrt{17} for x_{2}.

$$-x^{2}-8x+1=-\left(x-\left(-\left(\sqrt{17}+4\right)\right)\right)\left(x-\left(\sqrt{17}-4\right)\right)$$

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.

$$x ^ 2 +8x -1 = 0$$

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

$$r + s = -8 rs = -1$$

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.azureedge.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 = -1

$$(-4 - u) (-4 + u) = -1$$

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

$$16 - u^2 = -1$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = -1-16 = -17$$

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

$$u^2 = 17 u = \pm\sqrt{17} = \pm \sqrt{17}$$

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

$$r =-4 - \sqrt{17} = -8.123 s = -4 + \sqrt{17} = 0.123$$

Evaluate

$$1-8x-x^{2}$$