Solve -a+9/2a=10-(a+9)^2/4a

$$- \frac { a + 9 } { 2 a } = 10 - \frac { ( a + 9 ) ^ { 2 } } { 4 a }$$

Solve for a

$$a=3$$

$$a=21$$

Steps Using Factoring

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4a, the least common multiple of 2a,4a.

$$-2\left(a+9\right)=4a\times 10-\left(a+9\right)^{2}$$

Use the distributive property to multiply -2 by a+9.

$$-2a-18=4a\times 10-\left(a+9\right)^{2}$$

Multiply 4 and 10 to get 40.

$$-2a-18=40a-\left(a+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+9\right)^{2}.

$$-2a-18=40a-\left(a^{2}+18a+81\right)$$

To find the opposite of a^{2}+18a+81, find the opposite of each term.

$$-2a-18=40a-a^{2}-18a-81$$

Combine 40a and -18a to get 22a.

$$-2a-18=22a-a^{2}-81$$

Subtract 22a from both sides.

$$-2a-18-22a=-a^{2}-81$$

Combine -2a and -22a to get -24a.

$$-24a-18=-a^{2}-81$$

Add a^{2} to both sides.

$$-24a-18+a^{2}=-81$$

Add 81 to both sides.

$$-24a-18+a^{2}+81=0$$

Add -18 and 81 to get 63.

$$-24a+63+a^{2}=0$$

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

$$a^{2}-24a+63=0$$

To solve the equation, factor a^{2}-24a+63 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.

$$a+b=-24 ab=63$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 63.

$$-1,-63 -3,-21 -7,-9$$

Calculate the sum for each pair.

$$-1-63=-64 -3-21=-24 -7-9=-16$$

The solution is the pair that gives sum -24.

$$a=-21 b=-3$$

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

$$\left(a-21\right)\left(a-3\right)$$

To find equation solutions, solve a-21=0 and a-3=0.

$$a=21 a=3$$

Steps Using Factoring By Grouping

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4a, the least common multiple of 2a,4a.

$$-2\left(a+9\right)=4a\times 10-\left(a+9\right)^{2}$$

Use the distributive property to multiply -2 by a+9.

$$-2a-18=4a\times 10-\left(a+9\right)^{2}$$

Multiply 4 and 10 to get 40.

$$-2a-18=40a-\left(a+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+9\right)^{2}.

$$-2a-18=40a-\left(a^{2}+18a+81\right)$$

To find the opposite of a^{2}+18a+81, find the opposite of each term.

$$-2a-18=40a-a^{2}-18a-81$$

Combine 40a and -18a to get 22a.

$$-2a-18=22a-a^{2}-81$$

Subtract 22a from both sides.

$$-2a-18-22a=-a^{2}-81$$

Combine -2a and -22a to get -24a.

$$-24a-18=-a^{2}-81$$

Add a^{2} to both sides.

$$-24a-18+a^{2}=-81$$

Add 81 to both sides.

$$-24a-18+a^{2}+81=0$$

Add -18 and 81 to get 63.

$$-24a+63+a^{2}=0$$

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

$$a^{2}-24a+63=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba+63. To find a and b, set up a system to be solved.

$$a+b=-24 ab=1\times 63=63$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 63.

$$-1,-63 -3,-21 -7,-9$$

Calculate the sum for each pair.

$$-1-63=-64 -3-21=-24 -7-9=-16$$

The solution is the pair that gives sum -24.

$$a=-21 b=-3$$

Rewrite a^{2}-24a+63 as \left(a^{2}-21a\right)+\left(-3a+63\right).

$$\left(a^{2}-21a\right)+\left(-3a+63\right)$$

Factor out a in the first and -3 in the second group.

$$a\left(a-21\right)-3\left(a-21\right)$$

Factor out common term a-21 by using distributive property.

$$\left(a-21\right)\left(a-3\right)$$

To find equation solutions, solve a-21=0 and a-3=0.

$$a=21 a=3$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4a, the least common multiple of 2a,4a.

$$-2\left(a+9\right)=4a\times 10-\left(a+9\right)^{2}$$

Use the distributive property to multiply -2 by a+9.

$$-2a-18=4a\times 10-\left(a+9\right)^{2}$$

Multiply 4 and 10 to get 40.

$$-2a-18=40a-\left(a+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+9\right)^{2}.

$$-2a-18=40a-\left(a^{2}+18a+81\right)$$

To find the opposite of a^{2}+18a+81, find the opposite of each term.

$$-2a-18=40a-a^{2}-18a-81$$

Combine 40a and -18a to get 22a.

$$-2a-18=22a-a^{2}-81$$

Subtract 22a from both sides.

$$-2a-18-22a=-a^{2}-81$$

Combine -2a and -22a to get -24a.

$$-24a-18=-a^{2}-81$$

Add a^{2} to both sides.

$$-24a-18+a^{2}=-81$$

Add 81 to both sides.

$$-24a-18+a^{2}+81=0$$

Add -18 and 81 to get 63.

$$-24a+63+a^{2}=0$$

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.

$$a^{2}-24a+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -24 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 63}}{2}$$

Square -24.

$$a=\frac{-\left(-24\right)±\sqrt{576-4\times 63}}{2}$$

Multiply -4 times 63.

$$a=\frac{-\left(-24\right)±\sqrt{576-252}}{2}$$

Add 576 to -252.

$$a=\frac{-\left(-24\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$a=\frac{-\left(-24\right)±18}{2}$$

The opposite of -24 is 24.

$$a=\frac{24±18}{2}$$

Now solve the equation a=\frac{24±18}{2} when ± is plus. Add 24 to 18.

$$a=\frac{42}{2}$$

Divide 42 by 2.

$$a=21$$

Now solve the equation a=\frac{24±18}{2} when ± is minus. Subtract 18 from 24.

$$a=\frac{6}{2}$$

Divide 6 by 2.

$$a=3$$

The equation is now solved.

$$a=21 a=3$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4a, the least common multiple of 2a,4a.

$$-2\left(a+9\right)=4a\times 10-\left(a+9\right)^{2}$$

Use the distributive property to multiply -2 by a+9.

$$-2a-18=4a\times 10-\left(a+9\right)^{2}$$

Multiply 4 and 10 to get 40.

$$-2a-18=40a-\left(a+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+9\right)^{2}.

$$-2a-18=40a-\left(a^{2}+18a+81\right)$$

To find the opposite of a^{2}+18a+81, find the opposite of each term.

$$-2a-18=40a-a^{2}-18a-81$$

Combine 40a and -18a to get 22a.

$$-2a-18=22a-a^{2}-81$$

Subtract 22a from both sides.

$$-2a-18-22a=-a^{2}-81$$

Combine -2a and -22a to get -24a.

$$-24a-18=-a^{2}-81$$

Add a^{2} to both sides.

$$-24a-18+a^{2}=-81$$

Add 18 to both sides.

$$-24a+a^{2}=-81+18$$

Add -81 and 18 to get -63.

$$-24a+a^{2}=-63$$

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.

$$a^{2}-24a=-63$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-24a+\left(-12\right)^{2}=-63+\left(-12\right)^{2}$$

Square -12.

$$a^{2}-24a+144=-63+144$$

Add -63 to 144.

$$a^{2}-24a+144=81$$

Factor a^{2}-24a+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-12\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-12\right)^{2}}=\sqrt{81}$$

Simplify.

$$a-12=9 a-12=-9$$

Add 12 to both sides of the equation.

$$a=21 a=3$$