Solve 105+8x-x^2

$$105 + 8 x - x ^ { 2 }$$

Factor

$$-\left(x-15\right)\left(x+7\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+8x+105$$

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+105. To find a and b, set up a system to be solved.

$$a+b=8 ab=-105=-105$$

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 -105.

$$-1,105 -3,35 -5,21 -7,15$$

Calculate the sum for each pair.

$$-1+105=104 -3+35=32 -5+21=16 -7+15=8$$

The solution is the pair that gives sum 8.

$$a=15 b=-7$$

Rewrite -x^{2}+8x+105 as \left(-x^{2}+15x\right)+\left(-7x+105\right).

$$\left(-x^{2}+15x\right)+\left(-7x+105\right)$$

Factor out -x in the first and -7 in the second group.

$$-x\left(x-15\right)-7\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(-x-7\right)$$

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.

$$-x^{2}+8x+105=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\times 105}}{2\left(-1\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\times 105}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-8±\sqrt{64+4\times 105}}{2\left(-1\right)}$$

Multiply 4 times 105.

$$x=\frac{-8±\sqrt{64+420}}{2\left(-1\right)}$$

Add 64 to 420.

$$x=\frac{-8±\sqrt{484}}{2\left(-1\right)}$$

Take the square root of 484.

$$x=\frac{-8±22}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-8±22}{-2}$$

Now solve the equation x=\frac{-8±22}{-2} when ± is plus. Add -8 to 22.

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

Divide 14 by -2.

$$x=-7$$

Now solve the equation x=\frac{-8±22}{-2} when ± is minus. Subtract 22 from -8.

$$x=-\frac{30}{-2}$$

Divide -30 by -2.

$$x=15$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -7 for x_{1} and 15 for x_{2}.

$$-x^{2}+8x+105=-\left(x-\left(-7\right)\right)\left(x-15\right)$$

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

$$-x^{2}+8x+105=-\left(x+7\right)\left(x-15\right)$$

Evaluate

$$-\left(x-15\right)\left(x+7\right)$$