Solve x-left(2+xright)^2=-1.7

$$x- { \left(2+x \right) }^{ 2 } = -1.7$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{5}}{10}-1.5\approx -1.5+0.223606798i$$

$$x=-\frac{i\sqrt{5}}{10}-1.5\approx -1.5-0.223606798i$$

Steps Using the Quadratic Formula

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

$$x-\left(4+4x+x^{2}\right)=-1.7$$

To find the opposite of 4+4x+x^{2}, find the opposite of each term.

$$x-4-4x-x^{2}=-1.7$$

Combine x and -4x to get -3x.

$$-3x-4-x^{2}=-1.7$$

Add 1.7 to both sides.

$$-3x-4-x^{2}+1.7=0$$

Add -4 and 1.7 to get -2.3.

$$-3x-2.3-x^{2}=0$$

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.

$$-x^{2}-3x-2.3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -3 for b, and -2.3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\left(-2.3\right)}}{2\left(-1\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\left(-2.3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\left(-2.3\right)}}{2\left(-1\right)}$$

Multiply 4 times -2.3.

$$x=\frac{-\left(-3\right)±\sqrt{9-9.2}}{2\left(-1\right)}$$

Add 9 to -9.2.

$$x=\frac{-\left(-3\right)±\sqrt{-0.2}}{2\left(-1\right)}$$

Take the square root of -0.2.

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

The opposite of -3 is 3.

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

Multiply 2 times -1.

$$x=\frac{3±\frac{\sqrt{5}i}{5}}{-2}$$

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

$$x=\frac{\frac{\sqrt{5}i}{5}+3}{-2}$$

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

$$x=-\frac{\sqrt{5}i}{10}-\frac{3}{2}$$

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

$$x=\frac{-\frac{\sqrt{5}i}{5}+3}{-2}$$

Divide 3-\frac{i\sqrt{5}}{5} by -2.

$$x=\frac{\sqrt{5}i}{10}-\frac{3}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{5}i}{10}-\frac{3}{2} x=\frac{\sqrt{5}i}{10}-\frac{3}{2}$$

Steps for Completing the Square

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

$$x-\left(4+4x+x^{2}\right)=-1.7$$

To find the opposite of 4+4x+x^{2}, find the opposite of each term.

$$x-4-4x-x^{2}=-1.7$$

Combine x and -4x to get -3x.

$$-3x-4-x^{2}=-1.7$$

Add 4 to both sides.

$$-3x-x^{2}=-1.7+4$$

Add -1.7 and 4 to get 2.3.

$$-3x-x^{2}=2.3$$

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.

$$-x^{2}-3x=2.3$$

Divide both sides by -1.

$$\frac{-x^{2}-3x}{-1}=\frac{2.3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{3}{-1}\right)x=\frac{2.3}{-1}$$

Divide -3 by -1.

$$x^{2}+3x=\frac{2.3}{-1}$$

Divide 2.3 by -1.

$$x^{2}+3x=-2.3$$

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.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-2.3+\left(\frac{3}{2}\right)^{2}$$

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

$$x^{2}+3x+\frac{9}{4}=-2.3+\frac{9}{4}$$

Add -2.3 to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=-\frac{1}{20}$$

Factor x^{2}+3x+\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}.

$$\left(x+\frac{3}{2}\right)^{2}=-\frac{1}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{-\frac{1}{20}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{5}i}{10} x+\frac{3}{2}=-\frac{\sqrt{5}i}{10}$$

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

$$x=\frac{\sqrt{5}i}{10}-\frac{3}{2} x=-\frac{\sqrt{5}i}{10}-\frac{3}{2}$$