Solve 7=(a-2)^2/9-a

$$7 = \frac { ( a - 2 ) ^ { 2 } } { 9 - a }$$

Solve for a

$$a = \frac{7 \sqrt{5} - 3}{2} \approx 6.326237921$$

$$a=\frac{-7\sqrt{5}-3}{2}\approx -9.326237921$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 9 since division by zero is not defined. Multiply both sides of the equation by -a+9.

$$7\left(-a+9\right)=\left(a-2\right)^{2}$$

Use the distributive property to multiply 7 by -a+9.

$$-7a+63=\left(a-2\right)^{2}$$

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

$$-7a+63=a^{2}-4a+4$$

Subtract a^{2} from both sides.

$$-7a+63-a^{2}=-4a+4$$

Add 4a to both sides.

$$-7a+63-a^{2}+4a=4$$

Combine -7a and 4a to get -3a.

$$-3a+63-a^{2}=4$$

Subtract 4 from both sides.

$$-3a+63-a^{2}-4=0$$

Subtract 4 from 63 to get 59.

$$-3a+59-a^{2}=0$$

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.

$$-a^{2}-3a+59=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -3 for b, and 59 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 59}}{2\left(-1\right)}$$

Square -3.

$$a=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 59}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-\left(-3\right)±\sqrt{9+4\times 59}}{2\left(-1\right)}$$

Multiply 4 times 59.

$$a=\frac{-\left(-3\right)±\sqrt{9+236}}{2\left(-1\right)}$$

Add 9 to 236.

$$a=\frac{-\left(-3\right)±\sqrt{245}}{2\left(-1\right)}$$

Take the square root of 245.

$$a=\frac{-\left(-3\right)±7\sqrt{5}}{2\left(-1\right)}$$

The opposite of -3 is 3.

$$a=\frac{3±7\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{3±7\sqrt{5}}{-2}$$

Now solve the equation a=\frac{3±7\sqrt{5}}{-2} when ± is plus. Add 3 to 7\sqrt{5}.

$$a=\frac{7\sqrt{5}+3}{-2}$$

Divide 3+7\sqrt{5} by -2.

$$a=\frac{-7\sqrt{5}-3}{2}$$

Now solve the equation a=\frac{3±7\sqrt{5}}{-2} when ± is minus. Subtract 7\sqrt{5} from 3.

$$a=\frac{3-7\sqrt{5}}{-2}$$

Divide 3-7\sqrt{5} by -2.

$$a=\frac{7\sqrt{5}-3}{2}$$

The equation is now solved.

$$a=\frac{-7\sqrt{5}-3}{2} a=\frac{7\sqrt{5}-3}{2}$$

Steps for Completing the Square

Variable a cannot be equal to 9 since division by zero is not defined. Multiply both sides of the equation by -a+9.

$$7\left(-a+9\right)=\left(a-2\right)^{2}$$

Use the distributive property to multiply 7 by -a+9.

$$-7a+63=\left(a-2\right)^{2}$$

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

$$-7a+63=a^{2}-4a+4$$

Subtract a^{2} from both sides.

$$-7a+63-a^{2}=-4a+4$$

Add 4a to both sides.

$$-7a+63-a^{2}+4a=4$$

Combine -7a and 4a to get -3a.

$$-3a+63-a^{2}=4$$

Subtract 63 from both sides.

$$-3a-a^{2}=4-63$$

Subtract 63 from 4 to get -59.

$$-3a-a^{2}=-59$$

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.

$$-a^{2}-3a=-59$$

Divide both sides by -1.

$$\frac{-a^{2}-3a}{-1}=-\frac{59}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\left(-\frac{3}{-1}\right)a=-\frac{59}{-1}$$

Divide -3 by -1.

$$a^{2}+3a=-\frac{59}{-1}$$

Divide -59 by -1.

$$a^{2}+3a=59$$

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

$$a^{2}+3a+\left(\frac{3}{2}\right)^{2}=59+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+3a+\frac{9}{4}=59+\frac{9}{4}$$

Add 59 to \frac{9}{4}.

$$a^{2}+3a+\frac{9}{4}=\frac{245}{4}$$

Factor a^{2}+3a+\frac{9}{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}.

$$\left(a+\frac{3}{2}\right)^{2}=\frac{245}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{3}{2}\right)^{2}}=\sqrt{\frac{245}{4}}$$

Simplify.

$$a+\frac{3}{2}=\frac{7\sqrt{5}}{2} a+\frac{3}{2}=-\frac{7\sqrt{5}}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$a=\frac{7\sqrt{5}-3}{2} a=\frac{-7\sqrt{5}-3}{2}$$