Solve x^2+x=7x+16

$$x ^ { 2 } + x = 7 x + 16$$

Solve for x

$$x=-2$$

$$x=8$$

Steps Using Factoring

Subtract 7x from both sides.

$$x^{2}+x-7x=16$$

Combine x and -7x to get -6x.

$$x^{2}-6x=16$$

Subtract 16 from both sides.

$$x^{2}-6x-16=0$$

To solve the equation, factor x^{2}-6x-16 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-6 ab=-16$$

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 -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -6.

$$a=-8 b=2$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-8\right)\left(x+2\right)$$

To find equation solutions, solve x-8=0 and x+2=0.

$$x=8 x=-2$$

Steps Using Factoring By Grouping

Subtract 7x from both sides.

$$x^{2}+x-7x=16$$

Combine x and -7x to get -6x.

$$x^{2}-6x=16$$

Subtract 16 from both sides.

$$x^{2}-6x-16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-16\right)=-16$$

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 -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -6.

$$a=-8 b=2$$

Rewrite x^{2}-6x-16 as \left(x^{2}-8x\right)+\left(2x-16\right).

$$\left(x^{2}-8x\right)+\left(2x-16\right)$$

Factor out x in the first and 2 in the second group.

$$x\left(x-8\right)+2\left(x-8\right)$$

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

$$\left(x-8\right)\left(x+2\right)$$

To find equation solutions, solve x-8=0 and x+2=0.

$$x=8 x=-2$$

Steps Using the Quadratic Formula

Subtract 7x from both sides.

$$x^{2}+x-7x=16$$

Combine x and -7x to get -6x.

$$x^{2}-6x=16$$

Subtract 16 from both sides.

$$x^{2}-6x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-16\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-16\right)}}{2}$$

Multiply -4 times -16.

$$x=\frac{-\left(-6\right)±\sqrt{36+64}}{2}$$

Add 36 to 64.

$$x=\frac{-\left(-6\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-\left(-6\right)±10}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±10}{2}$$

Now solve the equation x=\frac{6±10}{2} when ± is plus. Add 6 to 10.

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

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{6±10}{2} when ± is minus. Subtract 10 from 6.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=8 x=-2$$

Steps for Completing the Square

Subtract 7x from both sides.

$$x^{2}+x-7x=16$$

Combine x and -7x to get -6x.

$$x^{2}-6x=16$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=16+9$$

Add 16 to 9.

$$x^{2}-6x+9=25$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-3=5 x-3=-5$$

Add 3 to both sides of the equation.

$$x=8 x=-2$$