Solve x^4-3.75x-1=x^4-3x^2-4

$${ x }^{ 4 } -3.75x-1 = { x }^{ 4 } -3 { x }^{ 2 } -4$$

Solve for x (complex solution)

$$x=\frac{5+i\sqrt{39}}{8}\approx 0.625+0.78062475i$$

$$x=\frac{-i\sqrt{39}+5}{8}\approx 0.625-0.78062475i$$

Steps Using the Quadratic Formula

Subtract x^{4} from both sides.

$$x^{4}-3.75x-1-x^{4}=-3x^{2}-4$$

Combine x^{4} and -x^{4} to get 0.

$$-3.75x-1=-3x^{2}-4$$

Add 3x^{2} to both sides.

$$-3.75x-1+3x^{2}=-4$$

Add 4 to both sides.

$$-3.75x-1+3x^{2}+4=0$$

Add -1 and 4 to get 3.

$$-3.75x+3+3x^{2}=0$$

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.

$$3x^{2}-3.75x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -3.75 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3.75\right)±\sqrt{\left(-3.75\right)^{2}-4\times 3\times 3}}{2\times 3}$$

Square -3.75 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-3.75\right)±\sqrt{14.0625-4\times 3\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-3.75\right)±\sqrt{14.0625-12\times 3}}{2\times 3}$$

Multiply -12 times 3.

$$x=\frac{-\left(-3.75\right)±\sqrt{14.0625-36}}{2\times 3}$$

Add 14.0625 to -36.

$$x=\frac{-\left(-3.75\right)±\sqrt{-21.9375}}{2\times 3}$$

Take the square root of -21.9375.

$$x=\frac{-\left(-3.75\right)±\frac{3\sqrt{39}i}{4}}{2\times 3}$$

The opposite of -3.75 is 3.75.

$$x=\frac{3.75±\frac{3\sqrt{39}i}{4}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{3.75±\frac{3\sqrt{39}i}{4}}{6}$$

Now solve the equation x=\frac{3.75±\frac{3\sqrt{39}i}{4}}{6} when ± is plus. Add 3.75 to \frac{3i\sqrt{39}}{4}.

$$x=\frac{15+3\sqrt{39}i}{4\times 6}$$

Divide \frac{15+3i\sqrt{39}}{4} by 6.

$$x=\frac{5+\sqrt{39}i}{8}$$

Now solve the equation x=\frac{3.75±\frac{3\sqrt{39}i}{4}}{6} when ± is minus. Subtract \frac{3i\sqrt{39}}{4} from 3.75.

$$x=\frac{-3\sqrt{39}i+15}{4\times 6}$$

Divide \frac{15-3i\sqrt{39}}{4} by 6.

$$x=\frac{-\sqrt{39}i+5}{8}$$

The equation is now solved.

$$x=\frac{5+\sqrt{39}i}{8} x=\frac{-\sqrt{39}i+5}{8}$$

Steps for Completing the Square

Subtract x^{4} from both sides.

$$x^{4}-3.75x-1-x^{4}=-3x^{2}-4$$

Combine x^{4} and -x^{4} to get 0.

$$-3.75x-1=-3x^{2}-4$$

Add 3x^{2} to both sides.

$$-3.75x-1+3x^{2}=-4$$

Add 1 to both sides.

$$-3.75x+3x^{2}=-4+1$$

Add -4 and 1 to get -3.

$$-3.75x+3x^{2}=-3$$

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.

$$3x^{2}-3.75x=-3$$

Divide both sides by 3.

$$\frac{3x^{2}-3.75x}{3}=-\frac{3}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{3.75}{3}\right)x=-\frac{3}{3}$$

Divide -3.75 by 3.

$$x^{2}-1.25x=-\frac{3}{3}$$

Divide -3 by 3.

$$x^{2}-1.25x=-1$$

Divide -1.25, the coefficient of the x term, by 2 to get -0.625. Then add the square of -0.625 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1.25x+\left(-0.625\right)^{2}=-1+\left(-0.625\right)^{2}$$

Square -0.625 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-1.25x+0.390625=-1+0.390625$$

Add -1 to 0.390625.

$$x^{2}-1.25x+0.390625=-0.609375$$

Factor x^{2}-1.25x+0.390625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.625\right)^{2}=-0.609375$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.625\right)^{2}}=\sqrt{-0.609375}$$

Simplify.

$$x-0.625=\frac{\sqrt{39}i}{8} x-0.625=-\frac{\sqrt{39}i}{8}$$

Add 0.625 to both sides of the equation.

$$x=\frac{5+\sqrt{39}i}{8} x=\frac{-\sqrt{39}i+5}{8}$$