Solve x^2-8x-1280

$${ x }^{ 2 } -8x-1280$$

Factor

$$\left(x-40\right)\left(x+32\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-1280. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\left(-1280\right)=-1280$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1280.

$$1,-1280 2,-640 4,-320 5,-256 8,-160 10,-128 16,-80 20,-64 32,-40$$

Calculate the sum for each pair.

$$1-1280=-1279 2-640=-638 4-320=-316 5-256=-251 8-160=-152 10-128=-118 16-80=-64 20-64=-44 32-40=-8$$

The solution is the pair that gives sum -8.

$$a=-40 b=32$$

Rewrite x^{2}-8x-1280 as \left(x^{2}-40x\right)+\left(32x-1280\right).

$$\left(x^{2}-40x\right)+\left(32x-1280\right)$$

Factor out x in the first and 32 in the second group.

$$x\left(x-40\right)+32\left(x-40\right)$$

Factor out common term x-40 by using distributive property.

$$\left(x-40\right)\left(x+32\right)$$

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.

$$x^{2}-8x-1280=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1280\right)}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1280\right)}}{2}$$

Multiply -4 times -1280.

$$x=\frac{-\left(-8\right)±\sqrt{64+5120}}{2}$$

Add 64 to 5120.

$$x=\frac{-\left(-8\right)±\sqrt{5184}}{2}$$

Take the square root of 5184.

$$x=\frac{-\left(-8\right)±72}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±72}{2}$$

Now solve the equation x=\frac{8±72}{2} when ± is plus. Add 8 to 72.

$$x=\frac{80}{2}$$

Divide 80 by 2.

$$x=40$$

Now solve the equation x=\frac{8±72}{2} when ± is minus. Subtract 72 from 8.

$$x=-\frac{64}{2}$$

Divide -64 by 2.

$$x=-32$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 40 for x_{1} and -32 for x_{2}.

$$x^{2}-8x-1280=\left(x-40\right)\left(x-\left(-32\right)\right)$$

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

$$x^{2}-8x-1280=\left(x-40\right)\left(x+32\right)$$

Evaluate

$$\left(x-40\right)\left(x+32\right)$$