Solve 7-6x-x^2

$$7-6x- { x }^{ 2 }$$

Factor

$$\left(1-x\right)\left(x+7\right)$$

Solution Steps

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

$$-x^{2}-6x+7$$

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=-6 ab=-7=-7$$

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. The only such pair is the system solution.

$$a=1 b=-7$$

Rewrite -x^{2}-6x+7 as \left(-x^{2}+x\right)+\left(-7x+7\right).

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

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

$$x\left(-x+1\right)+7\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(x+7\right)$$

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.

$$-x^{2}-6x+7=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 7}}{2\left(-1\right)}$$

Multiply 4 times 7.

$$x=\frac{-\left(-6\right)±\sqrt{36+28}}{2\left(-1\right)}$$

Add 36 to 28.

$$x=\frac{-\left(-6\right)±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$x=\frac{-\left(-6\right)±8}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±8}{-2}$$

Now solve the equation x=\frac{6±8}{-2} when ± is plus. Add 6 to 8.

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

Divide 14 by -2.

$$x=-7$$

Now solve the equation x=\frac{6±8}{-2} when ± is minus. Subtract 8 from 6.

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

Divide -2 by -2.

$$x=1$$

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 1 for x_{2}.

$$-x^{2}-6x+7=-\left(x-\left(-7\right)\right)\left(x-1\right)$$

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

$$-x^{2}-6x+7=-\left(x+7\right)\left(x-1\right)$$

Evaluate

$$\left(1-x\right)\left(x+7\right)$$