Solve 5x+6=-30-4x^2

$$5 x + 6 = - 30 - 4 x ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{551}i}{8}\approx -0.625+2.934173649i$$

$$x=\frac{-\sqrt{551}i-5}{8}\approx -0.625-2.934173649i$$

Steps Using the Quadratic Formula

Subtract -30 from both sides.

$$5x+6-\left(-30\right)=-4x^{2}$$

The opposite of -30 is 30.

$$5x+6+30=-4x^{2}$$

Add 4x^{2} to both sides.

$$5x+6+30+4x^{2}=0$$

Add 6 and 30 to get 36.

$$5x+36+4x^{2}=0$$

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.

$$4x^{2}+5x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 5 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 4\times 36}}{2\times 4}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 4\times 36}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-5±\sqrt{25-16\times 36}}{2\times 4}$$

Multiply -16 times 36.

$$x=\frac{-5±\sqrt{25-576}}{2\times 4}$$

Add 25 to -576.

$$x=\frac{-5±\sqrt{-551}}{2\times 4}$$

Take the square root of -551.

$$x=\frac{-5±\sqrt{551}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-5±\sqrt{551}i}{8}$$

Now solve the equation x=\frac{-5±\sqrt{551}i}{8} when ± is plus. Add -5 to i\sqrt{551}.

$$x=\frac{-5+\sqrt{551}i}{8}$$

Now solve the equation x=\frac{-5±\sqrt{551}i}{8} when ± is minus. Subtract i\sqrt{551} from -5.

$$x=\frac{-\sqrt{551}i-5}{8}$$

The equation is now solved.

$$x=\frac{-5+\sqrt{551}i}{8} x=\frac{-\sqrt{551}i-5}{8}$$

Steps for Completing the Square

Add 4x^{2} to both sides.

$$5x+6+4x^{2}=-30$$

Subtract 6 from both sides.

$$5x+4x^{2}=-30-6$$

Subtract 6 from -30 to get -36.

$$5x+4x^{2}=-36$$

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.

$$4x^{2}+5x=-36$$

Divide both sides by 4.

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

Dividing by 4 undoes the multiplication by 4.

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

Divide -36 by 4.

$$x^{2}+\frac{5}{4}x=-9$$

Divide \frac{5}{4}, the coefficient of the x term, by 2 to get \frac{5}{8}. Then add the square of \frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=-9+\left(\frac{5}{8}\right)^{2}$$

Square \frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=-9+\frac{25}{64}$$

Add -9 to \frac{25}{64}.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{551}{64}$$

Factor x^{2}+\frac{5}{4}x+\frac{25}{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}.

$$\left(x+\frac{5}{8}\right)^{2}=-\frac{551}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{8}\right)^{2}}=\sqrt{-\frac{551}{64}}$$

Simplify.

$$x+\frac{5}{8}=\frac{\sqrt{551}i}{8} x+\frac{5}{8}=-\frac{\sqrt{551}i}{8}$$

Subtract \frac{5}{8} from both sides of the equation.

$$x=\frac{-5+\sqrt{551}i}{8} x=\frac{-\sqrt{551}i-5}{8}$$