Solve x^2+64x-576

$$x ^ { 2 } + 64 x - 576$$

Factor

$$\left(x-8\right)\left(x+72\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-576. To find a and b, set up a system to be solved.

$$a+b=64 ab=1\left(-576\right)=-576$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -576.

$$-1,576 -2,288 -3,192 -4,144 -6,96 -8,72 -9,64 -12,48 -16,36 -18,32 -24,24$$

Calculate the sum for each pair.

$$-1+576=575 -2+288=286 -3+192=189 -4+144=140 -6+96=90 -8+72=64 -9+64=55 -12+48=36 -16+36=20 -18+32=14 -24+24=0$$

The solution is the pair that gives sum 64.

$$a=-8 b=72$$

Rewrite x^{2}+64x-576 as \left(x^{2}-8x\right)+\left(72x-576\right).

$$\left(x^{2}-8x\right)+\left(72x-576\right)$$

Factor out x in the first and 72 in the second group.

$$x\left(x-8\right)+72\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+72\right)$$

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.

$$x^{2}+64x-576=0$$

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.

$$x=\frac{-64±\sqrt{64^{2}-4\left(-576\right)}}{2}$$

Square 64.

$$x=\frac{-64±\sqrt{4096-4\left(-576\right)}}{2}$$

Multiply -4 times -576.

$$x=\frac{-64±\sqrt{4096+2304}}{2}$$

Add 4096 to 2304.

$$x=\frac{-64±\sqrt{6400}}{2}$$

Take the square root of 6400.

$$x=\frac{-64±80}{2}$$

Now solve the equation x=\frac{-64±80}{2} when ± is plus. Add -64 to 80.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{-64±80}{2} when ± is minus. Subtract 80 from -64.

$$x=-\frac{144}{2}$$

Divide -144 by 2.

$$x=-72$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 8 for x_{1} and -72 for x_{2}.

$$x^{2}+64x-576=\left(x-8\right)\left(x-\left(-72\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}+64x-576=\left(x-8\right)\left(x+72\right)$$

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.

$$x ^ 2 +64x -576 = 0$$

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

$$r + s = -64 rs = -576$$

Two numbers r and s sum up to -64 exactly when the average of the two numbers is \frac{1}{2}*-64 = -32. 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>

$$r = -32 - u s = -32 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -576

$$(-32 - u) (-32 + u) = -576$$

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

$$1024 - u^2 = -576$$

Simplify the expression by subtracting 1024 on both sides

$$-u^2 = -576-1024 = -1600$$

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

$$u^2 = 1600 u = \pm\sqrt{1600} = \pm 40$$

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

$$r =-32 - 40 = -72 s = -32 + 40 = 8$$

Evaluate

$$\left(x-8\right)\left(x+72\right)$$