Solve a^2-2a+3-(2^2-2*2+3)/a-2=9/4

$$\frac{ { a }^{ 2 } -2a+3-( { 2 }^{ 2 } -2 \cdot 2+3) }{ a-2 } = \frac{ 9 }{ 4 }$$

Solve for a

$$a = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

Steps Using Factoring By Grouping

Variable a cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 4\left(a-2\right), the least common multiple of a-2,4.

$$4\left(a^{2}-2a+3-\left(2^{2}-2\times 2+3\right)\right)=9\left(a-2\right)$$

Calculate 2 to the power of 2 and get 4.

$$4\left(a^{2}-2a+3-\left(4-2\times 2+3\right)\right)=9\left(a-2\right)$$

Multiply 2 and 2 to get 4.

$$4\left(a^{2}-2a+3-\left(4-4+3\right)\right)=9\left(a-2\right)$$

Subtract 4 from 4 to get 0.

$$4\left(a^{2}-2a+3-\left(0+3\right)\right)=9\left(a-2\right)$$

Add 0 and 3 to get 3.

$$4\left(a^{2}-2a+3-3\right)=9\left(a-2\right)$$

Subtract 3 from 3 to get 0.

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

Use the distributive property to multiply 4 by a^{2}-2a.

$$4a^{2}-8a=9\left(a-2\right)$$

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

$$4a^{2}-8a=9a-18$$

Subtract 9a from both sides.

$$4a^{2}-8a-9a=-18$$

Combine -8a and -9a to get -17a.

$$4a^{2}-17a=-18$$

Add 18 to both sides.

$$4a^{2}-17a+18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4a^{2}+aa+ba+18. To find a and b, set up a system to be solved.

$$a+b=-17 ab=4\times 18=72$$

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 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -17.

$$a=-9 b=-8$$

Rewrite 4a^{2}-17a+18 as \left(4a^{2}-9a\right)+\left(-8a+18\right).

$$\left(4a^{2}-9a\right)+\left(-8a+18\right)$$

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

$$a\left(4a-9\right)-2\left(4a-9\right)$$

Factor out common term 4a-9 by using distributive property.

$$\left(4a-9\right)\left(a-2\right)$$

To find equation solutions, solve 4a-9=0 and a-2=0.

$$a=\frac{9}{4} a=2$$

Variable a cannot be equal to 2.

$$a=\frac{9}{4}$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 4\left(a-2\right), the least common multiple of a-2,4.

$$4\left(a^{2}-2a+3-\left(2^{2}-2\times 2+3\right)\right)=9\left(a-2\right)$$

Calculate 2 to the power of 2 and get 4.

$$4\left(a^{2}-2a+3-\left(4-2\times 2+3\right)\right)=9\left(a-2\right)$$

Multiply 2 and 2 to get 4.

$$4\left(a^{2}-2a+3-\left(4-4+3\right)\right)=9\left(a-2\right)$$

Subtract 4 from 4 to get 0.

$$4\left(a^{2}-2a+3-\left(0+3\right)\right)=9\left(a-2\right)$$

Add 0 and 3 to get 3.

$$4\left(a^{2}-2a+3-3\right)=9\left(a-2\right)$$

Subtract 3 from 3 to get 0.

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

Use the distributive property to multiply 4 by a^{2}-2a.

$$4a^{2}-8a=9\left(a-2\right)$$

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

$$4a^{2}-8a=9a-18$$

Subtract 9a from both sides.

$$4a^{2}-8a-9a=-18$$

Combine -8a and -9a to get -17a.

$$4a^{2}-17a=-18$$

Add 18 to both sides.

$$4a^{2}-17a+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -17 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 4\times 18}}{2\times 4}$$

Square -17.

$$a=\frac{-\left(-17\right)±\sqrt{289-4\times 4\times 18}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-\left(-17\right)±\sqrt{289-16\times 18}}{2\times 4}$$

Multiply -16 times 18.

$$a=\frac{-\left(-17\right)±\sqrt{289-288}}{2\times 4}$$

Add 289 to -288.

$$a=\frac{-\left(-17\right)±\sqrt{1}}{2\times 4}$$

Take the square root of 1.

$$a=\frac{-\left(-17\right)±1}{2\times 4}$$

The opposite of -17 is 17.

$$a=\frac{17±1}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{17±1}{8}$$

Now solve the equation a=\frac{17±1}{8} when ± is plus. Add 17 to 1.

$$a=\frac{18}{8}$$

Reduce the fraction \frac{18}{8} to lowest terms by extracting and canceling out 2.

$$a=\frac{9}{4}$$

Now solve the equation a=\frac{17±1}{8} when ± is minus. Subtract 1 from 17.

$$a=\frac{16}{8}$$

Divide 16 by 8.

$$a=2$$

The equation is now solved.

$$a=\frac{9}{4} a=2$$

Variable a cannot be equal to 2.

$$a=\frac{9}{4}$$

Steps for Completing the Square

Variable a cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 4\left(a-2\right), the least common multiple of a-2,4.

$$4\left(a^{2}-2a+3-\left(2^{2}-2\times 2+3\right)\right)=9\left(a-2\right)$$

Calculate 2 to the power of 2 and get 4.

$$4\left(a^{2}-2a+3-\left(4-2\times 2+3\right)\right)=9\left(a-2\right)$$

Multiply 2 and 2 to get 4.

$$4\left(a^{2}-2a+3-\left(4-4+3\right)\right)=9\left(a-2\right)$$

Subtract 4 from 4 to get 0.

$$4\left(a^{2}-2a+3-\left(0+3\right)\right)=9\left(a-2\right)$$

Add 0 and 3 to get 3.

$$4\left(a^{2}-2a+3-3\right)=9\left(a-2\right)$$

Subtract 3 from 3 to get 0.

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

Use the distributive property to multiply 4 by a^{2}-2a.

$$4a^{2}-8a=9\left(a-2\right)$$

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

$$4a^{2}-8a=9a-18$$

Subtract 9a from both sides.

$$4a^{2}-8a-9a=-18$$

Combine -8a and -9a to get -17a.

$$4a^{2}-17a=-18$$

Divide both sides by 4.

$$\frac{4a^{2}-17a}{4}=-\frac{18}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$a^{2}-\frac{17}{4}a=-\frac{18}{4}$$

Reduce the fraction \frac{-18}{4} to lowest terms by extracting and canceling out 2.

$$a^{2}-\frac{17}{4}a=-\frac{9}{2}$$

Divide -\frac{17}{4}, the coefficient of the x term, by 2 to get -\frac{17}{8}. Then add the square of -\frac{17}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{17}{4}a+\left(-\frac{17}{8}\right)^{2}=-\frac{9}{2}+\left(-\frac{17}{8}\right)^{2}$$

Square -\frac{17}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{17}{4}a+\frac{289}{64}=-\frac{9}{2}+\frac{289}{64}$$

Add -\frac{9}{2} to \frac{289}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{17}{4}a+\frac{289}{64}=\frac{1}{64}$$

Factor a^{2}-\frac{17}{4}a+\frac{289}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{17}{8}\right)^{2}=\frac{1}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{17}{8}\right)^{2}}=\sqrt{\frac{1}{64}}$$

Simplify.

$$a-\frac{17}{8}=\frac{1}{8} a-\frac{17}{8}=-\frac{1}{8}$$

Add \frac{17}{8} to both sides of the equation.

$$a=\frac{9}{4} a=2$$

Variable a cannot be equal to 2.

$$a=\frac{9}{4}$$