Solve 56x^2+4x+85=0

$$56 { x }^{ 2 } +4x+85=0$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{1189}i}{28}\approx -0.035714286+1.231495689i$$

$$x=\frac{-\sqrt{1189}i-1}{28}\approx -0.035714286-1.231495689i$$

Steps Using the Quadratic Formula

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.

$$56x^{2}+4x+85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 56 for a, 4 for b, and 85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 56\times 85}}{2\times 56}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 56\times 85}}{2\times 56}$$

Multiply -4 times 56.

$$x=\frac{-4±\sqrt{16-224\times 85}}{2\times 56}$$

Multiply -224 times 85.

$$x=\frac{-4±\sqrt{16-19040}}{2\times 56}$$

Add 16 to -19040.

$$x=\frac{-4±\sqrt{-19024}}{2\times 56}$$

Take the square root of -19024.

$$x=\frac{-4±4\sqrt{1189}i}{2\times 56}$$

Multiply 2 times 56.

$$x=\frac{-4±4\sqrt{1189}i}{112}$$

Now solve the equation x=\frac{-4±4\sqrt{1189}i}{112} when ± is plus. Add -4 to 4i\sqrt{1189}.

$$x=\frac{-4+4\sqrt{1189}i}{112}$$

Divide -4+4i\sqrt{1189} by 112.

$$x=\frac{-1+\sqrt{1189}i}{28}$$

Now solve the equation x=\frac{-4±4\sqrt{1189}i}{112} when ± is minus. Subtract 4i\sqrt{1189} from -4.

$$x=\frac{-4\sqrt{1189}i-4}{112}$$

Divide -4-4i\sqrt{1189} by 112.

$$x=\frac{-\sqrt{1189}i-1}{28}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{1189}i}{28} x=\frac{-\sqrt{1189}i-1}{28}$$

Steps for Completing the Square

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.

$$56x^{2}+4x+85=0$$

Subtract 85 from both sides of the equation.

$$56x^{2}+4x+85-85=-85$$

Subtracting 85 from itself leaves 0.

$$56x^{2}+4x=-85$$

Divide both sides by 56.

$$\frac{56x^{2}+4x}{56}=-\frac{85}{56}$$

Dividing by 56 undoes the multiplication by 56.

$$x^{2}+\frac{4}{56}x=-\frac{85}{56}$$

Reduce the fraction \frac{4}{56} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{1}{14}x=-\frac{85}{56}$$

Divide \frac{1}{14}, the coefficient of the x term, by 2 to get \frac{1}{28}. Then add the square of \frac{1}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{14}x+\left(\frac{1}{28}\right)^{2}=-\frac{85}{56}+\left(\frac{1}{28}\right)^{2}$$

Square \frac{1}{28} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{14}x+\frac{1}{784}=-\frac{85}{56}+\frac{1}{784}$$

Add -\frac{85}{56} to \frac{1}{784} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{14}x+\frac{1}{784}=-\frac{1189}{784}$$

Factor x^{2}+\frac{1}{14}x+\frac{1}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{28}\right)^{2}=-\frac{1189}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{28}\right)^{2}}=\sqrt{-\frac{1189}{784}}$$

Simplify.

$$x+\frac{1}{28}=\frac{\sqrt{1189}i}{28} x+\frac{1}{28}=-\frac{\sqrt{1189}i}{28}$$

Subtract \frac{1}{28} from both sides of the equation.

$$x=\frac{-1+\sqrt{1189}i}{28} x=\frac{-\sqrt{1189}i-1}{28}$$