Solve x^2-49=4x^2+4x+1

$${ x }^{ 2 } -49=4 { x }^{ 2 } +4x+1$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{146}i-2}{3}\approx -0.666666667-4.027681991i$$

$$x=\frac{-2+\sqrt{146}i}{3}\approx -0.666666667+4.027681991i$$

Steps Using the Quadratic Formula

Subtract 4x^{2} from both sides.

$$x^{2}-49-4x^{2}=4x+1$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-49=4x+1$$

Subtract 4x from both sides.

$$-3x^{2}-49-4x=1$$

Subtract 1 from both sides.

$$-3x^{2}-49-4x-1=0$$

Subtract 1 from -49 to get -50.

$$-3x^{2}-50-4x=0$$

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.

$$-3x^{2}-4x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -4 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-3\right)\left(-50\right)}}{2\left(-3\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-3\right)\left(-50\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-4\right)±\sqrt{16+12\left(-50\right)}}{2\left(-3\right)}$$

Multiply 12 times -50.

$$x=\frac{-\left(-4\right)±\sqrt{16-600}}{2\left(-3\right)}$$

Add 16 to -600.

$$x=\frac{-\left(-4\right)±\sqrt{-584}}{2\left(-3\right)}$$

Take the square root of -584.

$$x=\frac{-\left(-4\right)±2\sqrt{146}i}{2\left(-3\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{146}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{4±2\sqrt{146}i}{-6}$$

Now solve the equation x=\frac{4±2\sqrt{146}i}{-6} when ± is plus. Add 4 to 2i\sqrt{146}.

$$x=\frac{4+2\sqrt{146}i}{-6}$$

Divide 4+2i\sqrt{146} by -6.

$$x=\frac{-\sqrt{146}i-2}{3}$$

Now solve the equation x=\frac{4±2\sqrt{146}i}{-6} when ± is minus. Subtract 2i\sqrt{146} from 4.

$$x=\frac{-2\sqrt{146}i+4}{-6}$$

Divide 4-2i\sqrt{146} by -6.

$$x=\frac{-2+\sqrt{146}i}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{146}i-2}{3} x=\frac{-2+\sqrt{146}i}{3}$$

Steps for Completing the Square

Subtract 4x^{2} from both sides.

$$x^{2}-49-4x^{2}=4x+1$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-49=4x+1$$

Subtract 4x from both sides.

$$-3x^{2}-49-4x=1$$

Add 49 to both sides.

$$-3x^{2}-4x=1+49$$

Add 1 and 49 to get 50.

$$-3x^{2}-4x=50$$

Divide both sides by -3.

$$\frac{-3x^{2}-4x}{-3}=\frac{50}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{4}{-3}\right)x=\frac{50}{-3}$$

Divide -4 by -3.

$$x^{2}+\frac{4}{3}x=\frac{50}{-3}$$

Divide 50 by -3.

$$x^{2}+\frac{4}{3}x=-\frac{50}{3}$$

Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{3}x+\left(\frac{2}{3}\right)^{2}=-\frac{50}{3}+\left(\frac{2}{3}\right)^{2}$$

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=-\frac{50}{3}+\frac{4}{9}$$

Add -\frac{50}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=-\frac{146}{9}$$

Factor x^{2}+\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{3}\right)^{2}=-\frac{146}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{-\frac{146}{9}}$$

Simplify.

$$x+\frac{2}{3}=\frac{\sqrt{146}i}{3} x+\frac{2}{3}=-\frac{\sqrt{146}i}{3}$$

Subtract \frac{2}{3} from both sides of the equation.

$$x=\frac{-2+\sqrt{146}i}{3} x=\frac{-\sqrt{146}i-2}{3}$$