Solve x^2+24x+64

$$x ^ { 2 } + 24 x + 64$$

Factor

$$\left(x-\left(-4\sqrt{5}-12\right)\right)\left(x-\left(4\sqrt{5}-12\right)\right)$$

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.

$$x^{2}+24x+64=0$$

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.

$$x=\frac{-24±\sqrt{24^{2}-4\times 64}}{2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 64}}{2}$$

Multiply -4 times 64.

$$x=\frac{-24±\sqrt{576-256}}{2}$$

Add 576 to -256.

$$x=\frac{-24±\sqrt{320}}{2}$$

Take the square root of 320.

$$x=\frac{-24±8\sqrt{5}}{2}$$

Now solve the equation x=\frac{-24±8\sqrt{5}}{2} when ± is plus. Add -24 to 8\sqrt{5}.

$$x=\frac{8\sqrt{5}-24}{2}$$

Divide -24+8\sqrt{5} by 2.

$$x=4\sqrt{5}-12$$

Now solve the equation x=\frac{-24±8\sqrt{5}}{2} when ± is minus. Subtract 8\sqrt{5} from -24.

$$x=\frac{-8\sqrt{5}-24}{2}$$

Divide -24-8\sqrt{5} by 2.

$$x=-4\sqrt{5}-12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -12+4\sqrt{5} for x_{1} and -12-4\sqrt{5} for x_{2}.

$$x^{2}+24x+64=\left(x-\left(4\sqrt{5}-12\right)\right)\left(x-\left(-4\sqrt{5}-12\right)\right)$$

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.

$$x ^ 2 +24x +64 = 0$$

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

$$r + s = -24 rs = 64$$

Two numbers r and s sum up to -24 exactly when the average of the two numbers is \frac{1}{2}*-24 = -12. 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>

$$r = -12 - u s = -12 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 64

$$(-12 - u) (-12 + u) = 64$$

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

$$144 - u^2 = 64$$

Simplify the expression by subtracting 144 on both sides

$$-u^2 = 64-144 = -80$$

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

$$u^2 = 80 u = \pm\sqrt{80} = \pm \sqrt{80}$$

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

$$r =-12 - \sqrt{80} = -20.944 s = -12 + \sqrt{80} = -3.056$$

Evaluate

$$x^{2}+24x+64$$