Solve 2(5x+2+x)=-(4+5x^2)

$$2 ( 5 x + 2 + x ) = - ( 4 + 5 x ^ { 2 } )$$

Solve for x (complex solution)

$$x=-\frac{6}{5}+\frac{2}{5}i=-1.2+0.4i$$

$$x=-\frac{6}{5}-\frac{2}{5}i=-1.2-0.4i$$

Steps Using the Quadratic Formula

Combine 5x and x to get 6x.

$$2\left(6x+2\right)=-\left(4+5x^{2}\right)$$

Use the distributive property to multiply 2 by 6x+2.

$$12x+4=-\left(4+5x^{2}\right)$$

To find the opposite of 4+5x^{2}, find the opposite of each term.

$$12x+4=-4-5x^{2}$$

Subtract -4 from both sides.

$$12x+4-\left(-4\right)=-5x^{2}$$

The opposite of -4 is 4.

$$12x+4+4=-5x^{2}$$

Combine 4 and 4 to get 2\times 4.

$$12x+2\times 4=-5x^{2}$$

Add 5x^{2} to both sides.

$$12x+2\times 4+5x^{2}=0$$

Multiply 2 and 4 to get 8.

$$12x+8+5x^{2}=0$$

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.

$$5x^{2}+12x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 12 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 5\times 8}}{2\times 5}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 5\times 8}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-12±\sqrt{144-20\times 8}}{2\times 5}$$

Multiply -20 times 8.

$$x=\frac{-12±\sqrt{144-160}}{2\times 5}$$

Add 144 to -160.

$$x=\frac{-12±\sqrt{-16}}{2\times 5}$$

Take the square root of -16.

$$x=\frac{-12±4i}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-12±4i}{10}$$

Now solve the equation x=\frac{-12±4i}{10} when ± is plus. Add -12 to 4i.

$$x=\frac{-12+4i}{10}$$

Divide -12+4i by 10.

$$x=-\frac{6}{5}+\frac{2}{5}i$$

Now solve the equation x=\frac{-12±4i}{10} when ± is minus. Subtract 4i from -12.

$$x=\frac{-12-4i}{10}$$

Divide -12-4i by 10.

$$x=-\frac{6}{5}-\frac{2}{5}i$$

The equation is now solved.

$$x=-\frac{6}{5}+\frac{2}{5}i x=-\frac{6}{5}-\frac{2}{5}i$$

Steps for Completing the Square

Combine 5x and x to get 6x.

$$2\left(6x+2\right)=-\left(4+5x^{2}\right)$$

Use the distributive property to multiply 2 by 6x+2.

$$12x+4=-\left(4+5x^{2}\right)$$

To find the opposite of 4+5x^{2}, find the opposite of each term.

$$12x+4=-4-5x^{2}$$

Add 5x^{2} to both sides.

$$12x+4+5x^{2}=-4$$

Subtract 4 from both sides.

$$12x+5x^{2}=-4-4$$

Subtract 4 from -4 to get -8.

$$12x+5x^{2}=-8$$

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.

$$5x^{2}+12x=-8$$

Divide both sides by 5.

$$\frac{5x^{2}+12x}{5}=-\frac{8}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{12}{5}x=-\frac{8}{5}$$

Divide \frac{12}{5}, the coefficient of the x term, by 2 to get \frac{6}{5}. Then add the square of \frac{6}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{12}{5}x+\left(\frac{6}{5}\right)^{2}=-\frac{8}{5}+\left(\frac{6}{5}\right)^{2}$$

Square \frac{6}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=-\frac{8}{5}+\frac{36}{25}$$

Add -\frac{8}{5} to \frac{36}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{12}{5}x+\frac{36}{25}=-\frac{4}{25}$$

Factor x^{2}+\frac{12}{5}x+\frac{36}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{6}{5}\right)^{2}=-\frac{4}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{6}{5}\right)^{2}}=\sqrt{-\frac{4}{25}}$$

Simplify.

$$x+\frac{6}{5}=\frac{2}{5}i x+\frac{6}{5}=-\frac{2}{5}i$$

Subtract \frac{6}{5} from both sides of the equation.

$$x=-\frac{6}{5}+\frac{2}{5}i x=-\frac{6}{5}-\frac{2}{5}i$$