Solve x+3=x+2/4x+1/4x

$$x+3= \frac{ x+2 }{ 4x } + \frac{ 1 }{ 4x }$$

Solve for x

$$x=-3$$

$$x=\frac{1}{4}=0.25$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x.

$$4xx+4x\times 3=x+2+1$$

Multiply x and x to get x^{2}.

$$4x^{2}+4x\times 3=x+2+1$$

Multiply 4 and 3 to get 12.

$$4x^{2}+12x=x+2+1$$

Add 2 and 1 to get 3.

$$4x^{2}+12x=x+3$$

Subtract x from both sides.

$$4x^{2}+12x-x=3$$

Combine 12x and -x to get 11x.

$$4x^{2}+11x=3$$

Subtract 3 from both sides.

$$4x^{2}+11x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 11 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 4\left(-3\right)}}{2\times 4}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 4\left(-3\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-11±\sqrt{121-16\left(-3\right)}}{2\times 4}$$

Multiply -16 times -3.

$$x=\frac{-11±\sqrt{121+48}}{2\times 4}$$

Add 121 to 48.

$$x=\frac{-11±\sqrt{169}}{2\times 4}$$

Take the square root of 169.

$$x=\frac{-11±13}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-11±13}{8}$$

Now solve the equation x=\frac{-11±13}{8} when ± is plus. Add -11 to 13.

$$x=\frac{2}{8}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{4}$$

Now solve the equation x=\frac{-11±13}{8} when ± is minus. Subtract 13 from -11.

$$x=-\frac{24}{8}$$

Divide -24 by 8.

$$x=-3$$

The equation is now solved.

$$x=\frac{1}{4} x=-3$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x.

$$4xx+4x\times 3=x+2+1$$

Multiply x and x to get x^{2}.

$$4x^{2}+4x\times 3=x+2+1$$

Multiply 4 and 3 to get 12.

$$4x^{2}+12x=x+2+1$$

Add 2 and 1 to get 3.

$$4x^{2}+12x=x+3$$

Subtract x from both sides.

$$4x^{2}+12x-x=3$$

Combine 12x and -x to get 11x.

$$4x^{2}+11x=3$$

Divide both sides by 4.

$$\frac{4x^{2}+11x}{4}=\frac{3}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{11}{4}x=\frac{3}{4}$$

Divide \frac{11}{4}, the coefficient of the x term, by 2 to get \frac{11}{8}. Then add the square of \frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{4}x+\left(\frac{11}{8}\right)^{2}=\frac{3}{4}+\left(\frac{11}{8}\right)^{2}$$

Square \frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{4}x+\frac{121}{64}=\frac{3}{4}+\frac{121}{64}$$

Add \frac{3}{4} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{4}x+\frac{121}{64}=\frac{169}{64}$$

Factor x^{2}+\frac{11}{4}x+\frac{121}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{8}\right)^{2}=\frac{169}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Simplify.

$$x+\frac{11}{8}=\frac{13}{8} x+\frac{11}{8}=-\frac{13}{8}$$

Subtract \frac{11}{8} from both sides of the equation.

$$x=\frac{1}{4} x=-3$$