Solve 3+left(x+2right)^2={left(2x+4/9right)}^2

$$3+ { \left(x+2 \right) }^{ 2 } = { \left( \frac{ 2x+4 }{ 9 } \right) }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{9\sqrt{231}i}{77}-2\approx -2+1.776469576i$$

$$x=-\frac{9\sqrt{231}i}{77}-2\approx -2-1.776469576i$$

Steps Using the Quadratic Formula

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

$$3+x^{2}+4x+4=\left(\frac{2x+4}{9}\right)^{2}$$

Add 3 and 4 to get 7.

$$7+x^{2}+4x=\left(\frac{2x+4}{9}\right)^{2}$$

To raise \frac{2x+4}{9} to a power, raise both numerator and denominator to the power and then divide.

$$7+x^{2}+4x=\frac{\left(2x+4\right)^{2}}{9^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+4\right)^{2}.

$$7+x^{2}+4x=\frac{4x^{2}+16x+16}{9^{2}}$$

Calculate 9 to the power of 2 and get 81.

$$7+x^{2}+4x=\frac{4x^{2}+16x+16}{81}$$

Divide each term of 4x^{2}+16x+16 by 81 to get \frac{4}{81}x^{2}+\frac{16}{81}x+\frac{16}{81}.

$$7+x^{2}+4x=\frac{4}{81}x^{2}+\frac{16}{81}x+\frac{16}{81}$$

Subtract \frac{4}{81}x^{2} from both sides.

$$7+x^{2}+4x-\frac{4}{81}x^{2}=\frac{16}{81}x+\frac{16}{81}$$

Combine x^{2} and -\frac{4}{81}x^{2} to get \frac{77}{81}x^{2}.

$$7+\frac{77}{81}x^{2}+4x=\frac{16}{81}x+\frac{16}{81}$$

Subtract \frac{16}{81}x from both sides.

$$7+\frac{77}{81}x^{2}+4x-\frac{16}{81}x=\frac{16}{81}$$

Combine 4x and -\frac{16}{81}x to get \frac{308}{81}x.

$$7+\frac{77}{81}x^{2}+\frac{308}{81}x=\frac{16}{81}$$

Subtract \frac{16}{81} from both sides.

$$7+\frac{77}{81}x^{2}+\frac{308}{81}x-\frac{16}{81}=0$$

Subtract \frac{16}{81} from 7 to get \frac{551}{81}.

$$\frac{551}{81}+\frac{77}{81}x^{2}+\frac{308}{81}x=0$$

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.

$$\frac{77}{81}x^{2}+\frac{308}{81}x+\frac{551}{81}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{77}{81} for a, \frac{308}{81} for b, and \frac{551}{81} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{308}{81}±\sqrt{\left(\frac{308}{81}\right)^{2}-4\times \frac{77}{81}\times \frac{551}{81}}}{2\times \frac{77}{81}}$$

Square \frac{308}{81} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{308}{81}±\sqrt{\frac{94864}{6561}-4\times \frac{77}{81}\times \frac{551}{81}}}{2\times \frac{77}{81}}$$

Multiply -4 times \frac{77}{81}.

$$x=\frac{-\frac{308}{81}±\sqrt{\frac{94864}{6561}-\frac{308}{81}\times \frac{551}{81}}}{2\times \frac{77}{81}}$$

Multiply -\frac{308}{81} times \frac{551}{81} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{308}{81}±\sqrt{\frac{94864-169708}{6561}}}{2\times \frac{77}{81}}$$

Add \frac{94864}{6561} to -\frac{169708}{6561} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{308}{81}±\sqrt{-\frac{308}{27}}}{2\times \frac{77}{81}}$$

Take the square root of -\frac{308}{27}.

$$x=\frac{-\frac{308}{81}±\frac{2\sqrt{231}i}{9}}{2\times \frac{77}{81}}$$

Multiply 2 times \frac{77}{81}.

$$x=\frac{-\frac{308}{81}±\frac{2\sqrt{231}i}{9}}{\frac{154}{81}}$$

Now solve the equation x=\frac{-\frac{308}{81}±\frac{2\sqrt{231}i}{9}}{\frac{154}{81}} when ± is plus. Add -\frac{308}{81} to \frac{2i\sqrt{231}}{9}.

$$x=\frac{\frac{2\sqrt{231}i}{9}-\frac{308}{81}}{\frac{154}{81}}$$

Divide -\frac{308}{81}+\frac{2i\sqrt{231}}{9} by \frac{154}{81} by multiplying -\frac{308}{81}+\frac{2i\sqrt{231}}{9} by the reciprocal of \frac{154}{81}.

$$x=\frac{9\sqrt{231}i}{77}-2$$

Now solve the equation x=\frac{-\frac{308}{81}±\frac{2\sqrt{231}i}{9}}{\frac{154}{81}} when ± is minus. Subtract \frac{2i\sqrt{231}}{9} from -\frac{308}{81}.

$$x=\frac{-\frac{2\sqrt{231}i}{9}-\frac{308}{81}}{\frac{154}{81}}$$

Divide -\frac{308}{81}-\frac{2i\sqrt{231}}{9} by \frac{154}{81} by multiplying -\frac{308}{81}-\frac{2i\sqrt{231}}{9} by the reciprocal of \frac{154}{81}.

$$x=-\frac{9\sqrt{231}i}{77}-2$$

The equation is now solved.

$$x=\frac{9\sqrt{231}i}{77}-2 x=-\frac{9\sqrt{231}i}{77}-2$$

Steps for Completing the Square

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

$$3+x^{2}+4x+4=\left(\frac{2x+4}{9}\right)^{2}$$

Add 3 and 4 to get 7.

$$7+x^{2}+4x=\left(\frac{2x+4}{9}\right)^{2}$$

To raise \frac{2x+4}{9} to a power, raise both numerator and denominator to the power and then divide.

$$7+x^{2}+4x=\frac{\left(2x+4\right)^{2}}{9^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+4\right)^{2}.

$$7+x^{2}+4x=\frac{4x^{2}+16x+16}{9^{2}}$$

Calculate 9 to the power of 2 and get 81.

$$7+x^{2}+4x=\frac{4x^{2}+16x+16}{81}$$

Divide each term of 4x^{2}+16x+16 by 81 to get \frac{4}{81}x^{2}+\frac{16}{81}x+\frac{16}{81}.

$$7+x^{2}+4x=\frac{4}{81}x^{2}+\frac{16}{81}x+\frac{16}{81}$$

Subtract \frac{4}{81}x^{2} from both sides.

$$7+x^{2}+4x-\frac{4}{81}x^{2}=\frac{16}{81}x+\frac{16}{81}$$

Combine x^{2} and -\frac{4}{81}x^{2} to get \frac{77}{81}x^{2}.

$$7+\frac{77}{81}x^{2}+4x=\frac{16}{81}x+\frac{16}{81}$$

Subtract \frac{16}{81}x from both sides.

$$7+\frac{77}{81}x^{2}+4x-\frac{16}{81}x=\frac{16}{81}$$

Combine 4x and -\frac{16}{81}x to get \frac{308}{81}x.

$$7+\frac{77}{81}x^{2}+\frac{308}{81}x=\frac{16}{81}$$

Subtract 7 from both sides.

$$\frac{77}{81}x^{2}+\frac{308}{81}x=\frac{16}{81}-7$$

Subtract 7 from \frac{16}{81} to get -\frac{551}{81}.

$$\frac{77}{81}x^{2}+\frac{308}{81}x=-\frac{551}{81}$$

Divide both sides of the equation by \frac{77}{81}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{77}{81}x^{2}+\frac{308}{81}x}{\frac{77}{81}}=-\frac{\frac{551}{81}}{\frac{77}{81}}$$

Dividing by \frac{77}{81} undoes the multiplication by \frac{77}{81}.

$$x^{2}+\frac{\frac{308}{81}}{\frac{77}{81}}x=-\frac{\frac{551}{81}}{\frac{77}{81}}$$

Divide \frac{308}{81} by \frac{77}{81} by multiplying \frac{308}{81} by the reciprocal of \frac{77}{81}.

$$x^{2}+4x=-\frac{\frac{551}{81}}{\frac{77}{81}}$$

Divide -\frac{551}{81} by \frac{77}{81} by multiplying -\frac{551}{81} by the reciprocal of \frac{77}{81}.

$$x^{2}+4x=-\frac{551}{77}$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=-\frac{551}{77}+2^{2}$$

Square 2.

$$x^{2}+4x+4=-\frac{551}{77}+4$$

Add -\frac{551}{77} to 4.

$$x^{2}+4x+4=-\frac{243}{77}$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=-\frac{243}{77}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{-\frac{243}{77}}$$

Simplify.

$$x+2=\frac{9\sqrt{231}i}{77} x+2=-\frac{9\sqrt{231}i}{77}$$

Subtract 2 from both sides of the equation.

$$x=\frac{9\sqrt{231}i}{77}-2 x=-\frac{9\sqrt{231}i}{77}-2$$