Solve 4+x=-16x^2+12x-4

$$4+x = -16 { x }^{ 2 } +12x-4$$

Solve for x (complex solution)

$$x=\frac{11+\sqrt{391}i}{32}\approx 0.34375+0.617928748i$$

$$x=\frac{-\sqrt{391}i+11}{32}\approx 0.34375-0.617928748i$$

Steps Using the Quadratic Formula

Add 16x^{2} to both sides.

$$4+x+16x^{2}=12x-4$$

Subtract 12x from both sides.

$$4+x+16x^{2}-12x=-4$$

Combine x and -12x to get -11x.

$$4-11x+16x^{2}=-4$$

Add 4 to both sides.

$$4-11x+16x^{2}+4=0$$

Add 4 and 4 to get 8.

$$8-11x+16x^{2}=0$$

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.

$$16x^{2}-11x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -11 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 16\times 8}}{2\times 16}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 16\times 8}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-11\right)±\sqrt{121-64\times 8}}{2\times 16}$$

Multiply -64 times 8.

$$x=\frac{-\left(-11\right)±\sqrt{121-512}}{2\times 16}$$

Add 121 to -512.

$$x=\frac{-\left(-11\right)±\sqrt{-391}}{2\times 16}$$

Take the square root of -391.

$$x=\frac{-\left(-11\right)±\sqrt{391}i}{2\times 16}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{391}i}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{11±\sqrt{391}i}{32}$$

Now solve the equation x=\frac{11±\sqrt{391}i}{32} when ± is plus. Add 11 to i\sqrt{391}.

$$x=\frac{11+\sqrt{391}i}{32}$$

Now solve the equation x=\frac{11±\sqrt{391}i}{32} when ± is minus. Subtract i\sqrt{391} from 11.

$$x=\frac{-\sqrt{391}i+11}{32}$$

The equation is now solved.

$$x=\frac{11+\sqrt{391}i}{32} x=\frac{-\sqrt{391}i+11}{32}$$

Steps for Completing the Square

Add 16x^{2} to both sides.

$$4+x+16x^{2}=12x-4$$

Subtract 12x from both sides.

$$4+x+16x^{2}-12x=-4$$

Combine x and -12x to get -11x.

$$4-11x+16x^{2}=-4$$

Subtract 4 from both sides.

$$-11x+16x^{2}=-4-4$$

Subtract 4 from -4 to get -8.

$$-11x+16x^{2}=-8$$

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.

$$16x^{2}-11x=-8$$

Divide both sides by 16.

$$\frac{16x^{2}-11x}{16}=-\frac{8}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}-\frac{11}{16}x=-\frac{8}{16}$$

Reduce the fraction \frac{-8}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{11}{16}x=-\frac{1}{2}$$

Divide -\frac{11}{16}, the coefficient of the x term, by 2 to get -\frac{11}{32}. Then add the square of -\frac{11}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{16}x+\left(-\frac{11}{32}\right)^{2}=-\frac{1}{2}+\left(-\frac{11}{32}\right)^{2}$$

Square -\frac{11}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{16}x+\frac{121}{1024}=-\frac{1}{2}+\frac{121}{1024}$$

Add -\frac{1}{2} to \frac{121}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{16}x+\frac{121}{1024}=-\frac{391}{1024}$$

Factor x^{2}-\frac{11}{16}x+\frac{121}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{32}\right)^{2}=-\frac{391}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{32}\right)^{2}}=\sqrt{-\frac{391}{1024}}$$

Simplify.

$$x-\frac{11}{32}=\frac{\sqrt{391}i}{32} x-\frac{11}{32}=-\frac{\sqrt{391}i}{32}$$

Add \frac{11}{32} to both sides of the equation.

$$x=\frac{11+\sqrt{391}i}{32} x=\frac{-\sqrt{391}i+11}{32}$$