Solve (-5x^2-x^2)=x+2+[2x+3]^2

$$( - 5 x ^ { 2 } - x ^ { 2 } ) = x + 2 + [ 2 x + 3 ] ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{271}i-13}{20}\approx -0.65-0.823103882i$$

$$x=\frac{-13+\sqrt{271}i}{20}\approx -0.65+0.823103882i$$

Steps Using the Quadratic Formula

Combine -5x^{2} and -x^{2} to get -6x^{2}.

$$-6x^{2}=x+2+\left(2x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$-6x^{2}=x+2+4x^{2}+12x+9$$

Combine x and 12x to get 13x.

$$-6x^{2}=13x+2+4x^{2}+9$$

Add 2 and 9 to get 11.

$$-6x^{2}=13x+11+4x^{2}$$

Subtract 13x from both sides.

$$-6x^{2}-13x=11+4x^{2}$$

Subtract 11 from both sides.

$$-6x^{2}-13x-11=4x^{2}$$

Subtract 4x^{2} from both sides.

$$-6x^{2}-13x-11-4x^{2}=0$$

Combine -6x^{2} and -4x^{2} to get -10x^{2}.

$$-10x^{2}-13x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, -13 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-10\right)\left(-11\right)}}{2\left(-10\right)}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-10\right)\left(-11\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-\left(-13\right)±\sqrt{169+40\left(-11\right)}}{2\left(-10\right)}$$

Multiply 40 times -11.

$$x=\frac{-\left(-13\right)±\sqrt{169-440}}{2\left(-10\right)}$$

Add 169 to -440.

$$x=\frac{-\left(-13\right)±\sqrt{-271}}{2\left(-10\right)}$$

Take the square root of -271.

$$x=\frac{-\left(-13\right)±\sqrt{271}i}{2\left(-10\right)}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{271}i}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{13±\sqrt{271}i}{-20}$$

Now solve the equation x=\frac{13±\sqrt{271}i}{-20} when ± is plus. Add 13 to i\sqrt{271}.

$$x=\frac{13+\sqrt{271}i}{-20}$$

Divide 13+i\sqrt{271} by -20.

$$x=\frac{-\sqrt{271}i-13}{20}$$

Now solve the equation x=\frac{13±\sqrt{271}i}{-20} when ± is minus. Subtract i\sqrt{271} from 13.

$$x=\frac{-\sqrt{271}i+13}{-20}$$

Divide 13-i\sqrt{271} by -20.

$$x=\frac{-13+\sqrt{271}i}{20}$$

The equation is now solved.

$$x=\frac{-\sqrt{271}i-13}{20} x=\frac{-13+\sqrt{271}i}{20}$$

Steps for Completing the Square

Combine -5x^{2} and -x^{2} to get -6x^{2}.

$$-6x^{2}=x+2+\left(2x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$-6x^{2}=x+2+4x^{2}+12x+9$$

Combine x and 12x to get 13x.

$$-6x^{2}=13x+2+4x^{2}+9$$

Add 2 and 9 to get 11.

$$-6x^{2}=13x+11+4x^{2}$$

Subtract 13x from both sides.

$$-6x^{2}-13x=11+4x^{2}$$

Subtract 4x^{2} from both sides.

$$-6x^{2}-13x-4x^{2}=11$$

Combine -6x^{2} and -4x^{2} to get -10x^{2}.

$$-10x^{2}-13x=11$$

Divide both sides by -10.

$$\frac{-10x^{2}-13x}{-10}=\frac{11}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\left(-\frac{13}{-10}\right)x=\frac{11}{-10}$$

Divide -13 by -10.

$$x^{2}+\frac{13}{10}x=\frac{11}{-10}$$

Divide 11 by -10.

$$x^{2}+\frac{13}{10}x=-\frac{11}{10}$$

Divide \frac{13}{10}, the coefficient of the x term, by 2 to get \frac{13}{20}. Then add the square of \frac{13}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{10}x+\left(\frac{13}{20}\right)^{2}=-\frac{11}{10}+\left(\frac{13}{20}\right)^{2}$$

Square \frac{13}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{10}x+\frac{169}{400}=-\frac{11}{10}+\frac{169}{400}$$

Add -\frac{11}{10} to \frac{169}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{13}{10}x+\frac{169}{400}=-\frac{271}{400}$$

Factor x^{2}+\frac{13}{10}x+\frac{169}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{20}\right)^{2}=-\frac{271}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{20}\right)^{2}}=\sqrt{-\frac{271}{400}}$$

Simplify.

$$x+\frac{13}{20}=\frac{\sqrt{271}i}{20} x+\frac{13}{20}=-\frac{\sqrt{271}i}{20}$$

Subtract \frac{13}{20} from both sides of the equation.

$$x=\frac{-13+\sqrt{271}i}{20} x=\frac{-\sqrt{271}i-13}{20}$$