Solve x^2+16x+57=

$$x ^ { 2 } + 16 x + 57 =$$

Factor

$$\left(x-\left(-\sqrt{7}-8\right)\right)\left(x-\left(\sqrt{7}-8\right)\right)$$

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.

$$x^{2}+16x+57=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\times 57}}{2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 57}}{2}$$

Multiply -4 times 57.

$$x=\frac{-16±\sqrt{256-228}}{2}$$

Add 256 to -228.

$$x=\frac{-16±\sqrt{28}}{2}$$

Take the square root of 28.

$$x=\frac{-16±2\sqrt{7}}{2}$$

Now solve the equation x=\frac{-16±2\sqrt{7}}{2} when ± is plus. Add -16 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-16}{2}$$

Divide -16+2\sqrt{7} by 2.

$$x=\sqrt{7}-8$$

Now solve the equation x=\frac{-16±2\sqrt{7}}{2} when ± is minus. Subtract 2\sqrt{7} from -16.

$$x=\frac{-2\sqrt{7}-16}{2}$$

Divide -16-2\sqrt{7} by 2.

$$x=-\sqrt{7}-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -8+\sqrt{7} for x_{1} and -8-\sqrt{7} for x_{2}.

$$x^{2}+16x+57=\left(x-\left(\sqrt{7}-8\right)\right)\left(x-\left(-\sqrt{7}-8\right)\right)$$

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.

$$x ^ 2 +16x +57 = 0$$

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

$$r + s = -16 rs = 57$$

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 = 57

$$(-8 - u) (-8 + u) = 57$$

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

$$64 - u^2 = 57$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = 57-64 = -7$$

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

$$u^2 = 7 u = \pm\sqrt{7} = \pm \sqrt{7}$$

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

$$r =-8 - \sqrt{7} = -10.646 s = -8 + \sqrt{7} = -5.354$$

Evaluate

$$x^{2}+16x+57$$