Solve 147x^2+2x+7=0

$$147 { x }^{ 2 } +2x+7=0$$

Solve for x (complex solution)

$$x=\frac{-1+2\sqrt{257}i}{147}\approx -0.006802721+0.218111831i$$

$$x=\frac{-2\sqrt{257}i-1}{147}\approx -0.006802721-0.218111831i$$

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.

$$147x^{2}+2x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 147 for a, 2 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 147\times 7}}{2\times 147}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 147\times 7}}{2\times 147}$$

Multiply -4 times 147.

$$x=\frac{-2±\sqrt{4-588\times 7}}{2\times 147}$$

Multiply -588 times 7.

$$x=\frac{-2±\sqrt{4-4116}}{2\times 147}$$

Add 4 to -4116.

$$x=\frac{-2±\sqrt{-4112}}{2\times 147}$$

Take the square root of -4112.

$$x=\frac{-2±4\sqrt{257}i}{2\times 147}$$

Multiply 2 times 147.

$$x=\frac{-2±4\sqrt{257}i}{294}$$

Now solve the equation x=\frac{-2±4\sqrt{257}i}{294} when ± is plus. Add -2 to 4i\sqrt{257}.

$$x=\frac{-2+4\sqrt{257}i}{294}$$

Divide -2+4i\sqrt{257} by 294.

$$x=\frac{-1+2\sqrt{257}i}{147}$$

Now solve the equation x=\frac{-2±4\sqrt{257}i}{294} when ± is minus. Subtract 4i\sqrt{257} from -2.

$$x=\frac{-4\sqrt{257}i-2}{294}$$

Divide -2-4i\sqrt{257} by 294.

$$x=\frac{-2\sqrt{257}i-1}{147}$$

The equation is now solved.

$$x=\frac{-1+2\sqrt{257}i}{147} x=\frac{-2\sqrt{257}i-1}{147}$$

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.

$$147x^{2}+2x+7=0$$

Subtract 7 from both sides of the equation.

$$147x^{2}+2x+7-7=-7$$

Subtracting 7 from itself leaves 0.

$$147x^{2}+2x=-7$$

Divide both sides by 147.

$$\frac{147x^{2}+2x}{147}=-\frac{7}{147}$$

Dividing by 147 undoes the multiplication by 147.

$$x^{2}+\frac{2}{147}x=-\frac{7}{147}$$

Reduce the fraction \frac{-7}{147} to lowest terms by extracting and canceling out 7.

$$x^{2}+\frac{2}{147}x=-\frac{1}{21}$$

Divide \frac{2}{147}, the coefficient of the x term, by 2 to get \frac{1}{147}. Then add the square of \frac{1}{147} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{147}x+\left(\frac{1}{147}\right)^{2}=-\frac{1}{21}+\left(\frac{1}{147}\right)^{2}$$

Square \frac{1}{147} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{147}x+\frac{1}{21609}=-\frac{1}{21}+\frac{1}{21609}$$

Add -\frac{1}{21} to \frac{1}{21609} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{147}x+\frac{1}{21609}=-\frac{1028}{21609}$$

Factor x^{2}+\frac{2}{147}x+\frac{1}{21609}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{147}\right)^{2}=-\frac{1028}{21609}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{147}\right)^{2}}=\sqrt{-\frac{1028}{21609}}$$

Simplify.

$$x+\frac{1}{147}=\frac{2\sqrt{257}i}{147} x+\frac{1}{147}=-\frac{2\sqrt{257}i}{147}$$

Subtract \frac{1}{147} from both sides of the equation.

$$x=\frac{-1+2\sqrt{257}i}{147} x=\frac{-2\sqrt{257}i-1}{147}$$