Solve 7x^2+4x+1568=0

$$7 { x }^{ 2 } +4x+1568 = 0$$

Solve for x (complex solution)

$$x=\frac{-2+2\sqrt{2743}i}{7}\approx -0.285714286+14.963902143i$$

$$x=\frac{-2\sqrt{2743}i-2}{7}\approx -0.285714286-14.963902143i$$

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.

$$7x^{2}+4x+1568=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 4 for b, and 1568 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 7\times 1568}}{2\times 7}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 7\times 1568}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-4±\sqrt{16-28\times 1568}}{2\times 7}$$

Multiply -28 times 1568.

$$x=\frac{-4±\sqrt{16-43904}}{2\times 7}$$

Add 16 to -43904.

$$x=\frac{-4±\sqrt{-43888}}{2\times 7}$$

Take the square root of -43888.

$$x=\frac{-4±4\sqrt{2743}i}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-4±4\sqrt{2743}i}{14}$$

Now solve the equation x=\frac{-4±4\sqrt{2743}i}{14} when ± is plus. Add -4 to 4i\sqrt{2743}.

$$x=\frac{-4+4\sqrt{2743}i}{14}$$

Divide -4+4i\sqrt{2743} by 14.

$$x=\frac{-2+2\sqrt{2743}i}{7}$$

Now solve the equation x=\frac{-4±4\sqrt{2743}i}{14} when ± is minus. Subtract 4i\sqrt{2743} from -4.

$$x=\frac{-4\sqrt{2743}i-4}{14}$$

Divide -4-4i\sqrt{2743} by 14.

$$x=\frac{-2\sqrt{2743}i-2}{7}$$

The equation is now solved.

$$x=\frac{-2+2\sqrt{2743}i}{7} x=\frac{-2\sqrt{2743}i-2}{7}$$

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.

$$7x^{2}+4x+1568=0$$

Subtract 1568 from both sides of the equation.

$$7x^{2}+4x+1568-1568=-1568$$

Subtracting 1568 from itself leaves 0.

$$7x^{2}+4x=-1568$$

Divide both sides by 7.

$$\frac{7x^{2}+4x}{7}=-\frac{1568}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{4}{7}x=-\frac{1568}{7}$$

Divide -1568 by 7.

$$x^{2}+\frac{4}{7}x=-224$$

Divide \frac{4}{7}, the coefficient of the x term, by 2 to get \frac{2}{7}. Then add the square of \frac{2}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{7}x+\left(\frac{2}{7}\right)^{2}=-224+\left(\frac{2}{7}\right)^{2}$$

Square \frac{2}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=-224+\frac{4}{49}$$

Add -224 to \frac{4}{49}.

$$x^{2}+\frac{4}{7}x+\frac{4}{49}=-\frac{10972}{49}$$

Factor x^{2}+\frac{4}{7}x+\frac{4}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{7}\right)^{2}=-\frac{10972}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{7}\right)^{2}}=\sqrt{-\frac{10972}{49}}$$

Simplify.

$$x+\frac{2}{7}=\frac{2\sqrt{2743}i}{7} x+\frac{2}{7}=-\frac{2\sqrt{2743}i}{7}$$

Subtract \frac{2}{7} from both sides of the equation.

$$x=\frac{-2+2\sqrt{2743}i}{7} x=\frac{-2\sqrt{2743}i-2}{7}$$