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

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

Factor

$$-\left(x-\left(4-\sqrt{14}\right)\right)\left(x-\left(\sqrt{14}+4\right)\right)$$

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.

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

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.

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

Square 8.

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

Multiply -4 times -1.

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

Multiply 4 times -2.

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

Add 64 to -8.

$$x=\frac{-8±\sqrt{56}}{2\left(-1\right)}$$

Take the square root of 56.

$$x=\frac{-8±2\sqrt{14}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-8±2\sqrt{14}}{-2}$$

Now solve the equation x=\frac{-8±2\sqrt{14}}{-2} when ± is plus. Add -8 to 2\sqrt{14}.

$$x=\frac{2\sqrt{14}-8}{-2}$$

Divide -8+2\sqrt{14} by -2.

$$x=4-\sqrt{14}$$

Now solve the equation x=\frac{-8±2\sqrt{14}}{-2} when ± is minus. Subtract 2\sqrt{14} from -8.

$$x=\frac{-2\sqrt{14}-8}{-2}$$

Divide -8-2\sqrt{14} by -2.

$$x=\sqrt{14}+4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4-\sqrt{14} for x_{1} and 4+\sqrt{14} for x_{2}.

$$-x^{2}+8x-2=-\left(x-\left(4-\sqrt{14}\right)\right)\left(x-\left(\sqrt{14}+4\right)\right)$$

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.

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

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

$$r + s = 8 rs = 2$$

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 = 2

$$(4 - u) (4 + u) = 2$$

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

$$16 - u^2 = 2$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = 2-16 = -14$$

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

$$u^2 = 14 u = \pm\sqrt{14} = \pm \sqrt{14}$$

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

$$r =4 - \sqrt{14} = 0.258 s = 4 + \sqrt{14} = 7.742$$

Evaluate

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