Solve a^2-5a=18+2a

$$a ^ { 2 } - 5 a = 18 + 2 a$$

Solve for a

$$a=-2$$

$$a=9$$

Steps Using Factoring

Subtract 18 from both sides.

$$a^{2}-5a-18=2a$$

Subtract 2a from both sides.

$$a^{2}-5a-18-2a=0$$

Combine -5a and -2a to get -7a.

$$a^{2}-7a-18=0$$

To solve the equation, factor a^{2}-7a-18 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). To find a and b, set up a system to be solved.

$$a+b=-7 ab=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -7.

$$a=-9 b=2$$

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

$$\left(a-9\right)\left(a+2\right)$$

To find equation solutions, solve a-9=0 and a+2=0.

$$a=9 a=-2$$

Steps Using Factoring By Grouping

Subtract 18 from both sides.

$$a^{2}-5a-18=2a$$

Subtract 2a from both sides.

$$a^{2}-5a-18-2a=0$$

Combine -5a and -2a to get -7a.

$$a^{2}-7a-18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba-18. To find a and b, set up a system to be solved.

$$a+b=-7 ab=1\left(-18\right)=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -7.

$$a=-9 b=2$$

Rewrite a^{2}-7a-18 as \left(a^{2}-9a\right)+\left(2a-18\right).

$$\left(a^{2}-9a\right)+\left(2a-18\right)$$

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

$$a\left(a-9\right)+2\left(a-9\right)$$

Factor out common term a-9 by using distributive property.

$$\left(a-9\right)\left(a+2\right)$$

To find equation solutions, solve a-9=0 and a+2=0.

$$a=9 a=-2$$

Steps Using the Quadratic Formula

Subtract 18 from both sides.

$$a^{2}-5a-18=2a$$

Subtract 2a from both sides.

$$a^{2}-5a-18-2a=0$$

Combine -5a and -2a to get -7a.

$$a^{2}-7a-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-18\right)}}{2}$$

Square -7.

$$a=\frac{-\left(-7\right)±\sqrt{49-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$a=\frac{-\left(-7\right)±\sqrt{49+72}}{2}$$

Add 49 to 72.

$$a=\frac{-\left(-7\right)±\sqrt{121}}{2}$$

Take the square root of 121.

$$a=\frac{-\left(-7\right)±11}{2}$$

The opposite of -7 is 7.

$$a=\frac{7±11}{2}$$

Now solve the equation a=\frac{7±11}{2} when ± is plus. Add 7 to 11.

$$a=\frac{18}{2}$$

Divide 18 by 2.

$$a=9$$

Now solve the equation a=\frac{7±11}{2} when ± is minus. Subtract 11 from 7.

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

Divide -4 by 2.

$$a=-2$$

The equation is now solved.

$$a=9 a=-2$$

Steps for Completing the Square

Subtract 2a from both sides.

$$a^{2}-5a-2a=18$$

Combine -5a and -2a to get -7a.

$$a^{2}-7a=18$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-7a+\left(-\frac{7}{2}\right)^{2}=18+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-7a+\frac{49}{4}=18+\frac{49}{4}$$

Add 18 to \frac{49}{4}.

$$a^{2}-7a+\frac{49}{4}=\frac{121}{4}$$

Factor a^{2}-7a+\frac{49}{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}.

$$\left(a-\frac{7}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{7}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$a-\frac{7}{2}=\frac{11}{2} a-\frac{7}{2}=-\frac{11}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$a=9 a=-2$$