Solve 2x-(x^2-7x)-8=0

$$2 x - ( x ^ { 2 } - 7 x ) - 8 = 0$$

Solve for x

$$x=1$$

$$x=8$$

Steps Using Factoring By Grouping

To find the opposite of x^{2}-7x, find the opposite of each term.

$$2x-x^{2}+7x-8=0$$

Combine 2x and 7x to get 9x.

$$9x-x^{2}-8=0$$

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

$$-x^{2}+9x-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-8. To find a and b, set up a system to be solved.

$$a+b=9 ab=-\left(-8\right)=8$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 9.

$$a=8 b=1$$

Rewrite -x^{2}+9x-8 as \left(-x^{2}+8x\right)+\left(x-8\right).

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

Factor out -x in -x^{2}+8x.

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

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

$$\left(x-8\right)\left(-x+1\right)$$

To find equation solutions, solve x-8=0 and -x+1=0.

$$x=8 x=1$$

Steps Using the Quadratic Formula

To find the opposite of x^{2}-7x, find the opposite of each term.

$$2x-x^{2}+7x-8=0$$

Combine 2x and 7x to get 9x.

$$9x-x^{2}-8=0$$

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.

$$-x^{2}+9x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\left(-8\right)}}{2\left(-1\right)}$$

Multiply 4 times -8.

$$x=\frac{-9±\sqrt{81-32}}{2\left(-1\right)}$$

Add 81 to -32.

$$x=\frac{-9±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-9±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±7}{-2}$$

Now solve the equation x=\frac{-9±7}{-2} when ± is plus. Add -9 to 7.

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

Divide -2 by -2.

$$x=1$$

Now solve the equation x=\frac{-9±7}{-2} when ± is minus. Subtract 7 from -9.

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

Divide -16 by -2.

$$x=8$$

The equation is now solved.

$$x=1 x=8$$

Steps for Completing the Square

To find the opposite of x^{2}-7x, find the opposite of each term.

$$2x-x^{2}+7x-8=0$$

Combine 2x and 7x to get 9x.

$$9x-x^{2}-8=0$$

Add 8 to both sides. Anything plus zero gives itself.

$$9x-x^{2}=8$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+9x=8$$

Divide both sides by -1.

$$\frac{-x^{2}+9x}{-1}=\frac{8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{9}{-1}x=\frac{8}{-1}$$

Divide 9 by -1.

$$x^{2}-9x=\frac{8}{-1}$$

Divide 8 by -1.

$$x^{2}-9x=-8$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-8+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-8+\frac{81}{4}$$

Add -8 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{49}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{7}{2} x-\frac{9}{2}=-\frac{7}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=8 x=1$$