Solve 4/3p+1/12=2/p

$$\frac { 4 } { 3 } p + \frac { 1 } { 12 } = \frac { 2 } { p }$$

Solve for p

$$p = \frac{\sqrt{1537} - 1}{32} \approx 1.193893486$$

$$p=\frac{-\sqrt{1537}-1}{32}\approx -1.256393486$$

Steps Using the Quadratic Formula

Variable p cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12p, the least common multiple of 3,12,p.

$$\frac{4}{3}p\times 12p+12p\times \frac{1}{12}=12\times 2$$

Express \frac{4}{3}\times 12 as a single fraction.

$$\frac{4\times 12}{3}pp+12p\times \frac{1}{12}=12\times 2$$

Multiply 4 and 12 to get 48.

$$\frac{48}{3}pp+12p\times \frac{1}{12}=12\times 2$$

Divide 48 by 3 to get 16.

$$16pp+12p\times \frac{1}{12}=12\times 2$$

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

$$16p^{2}+12p\times \frac{1}{12}=12\times 2$$

Cancel out 12 and 12.

$$16p^{2}+p=12\times 2$$

Multiply 12 and 2 to get 24.

$$16p^{2}+p=24$$

Subtract 24 from both sides.

$$16p^{2}+p-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 1 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-1±\sqrt{1^{2}-4\times 16\left(-24\right)}}{2\times 16}$$

Square 1.

$$p=\frac{-1±\sqrt{1-4\times 16\left(-24\right)}}{2\times 16}$$

Multiply -4 times 16.

$$p=\frac{-1±\sqrt{1-64\left(-24\right)}}{2\times 16}$$

Multiply -64 times -24.

$$p=\frac{-1±\sqrt{1+1536}}{2\times 16}$$

Add 1 to 1536.

$$p=\frac{-1±\sqrt{1537}}{2\times 16}$$

Multiply 2 times 16.

$$p=\frac{-1±\sqrt{1537}}{32}$$

Now solve the equation p=\frac{-1±\sqrt{1537}}{32} when ± is plus. Add -1 to \sqrt{1537}.

$$p=\frac{\sqrt{1537}-1}{32}$$

Now solve the equation p=\frac{-1±\sqrt{1537}}{32} when ± is minus. Subtract \sqrt{1537} from -1.

$$p=\frac{-\sqrt{1537}-1}{32}$$

The equation is now solved.

$$p=\frac{\sqrt{1537}-1}{32} p=\frac{-\sqrt{1537}-1}{32}$$

Steps for Completing the Square

Variable p cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12p, the least common multiple of 3,12,p.

$$\frac{4}{3}p\times 12p+12p\times \frac{1}{12}=12\times 2$$

Express \frac{4}{3}\times 12 as a single fraction.

$$\frac{4\times 12}{3}pp+12p\times \frac{1}{12}=12\times 2$$

Multiply 4 and 12 to get 48.

$$\frac{48}{3}pp+12p\times \frac{1}{12}=12\times 2$$

Divide 48 by 3 to get 16.

$$16pp+12p\times \frac{1}{12}=12\times 2$$

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

$$16p^{2}+12p\times \frac{1}{12}=12\times 2$$

Cancel out 12 and 12.

$$16p^{2}+p=12\times 2$$

Multiply 12 and 2 to get 24.

$$16p^{2}+p=24$$

Divide both sides by 16.

$$\frac{16p^{2}+p}{16}=\frac{24}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$p^{2}+\frac{1}{16}p=\frac{24}{16}$$

Reduce the fraction \frac{24}{16} to lowest terms by extracting and canceling out 8.

$$p^{2}+\frac{1}{16}p=\frac{3}{2}$$

Divide \frac{1}{16}, the coefficient of the x term, by 2 to get \frac{1}{32}. Then add the square of \frac{1}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+\frac{1}{16}p+\left(\frac{1}{32}\right)^{2}=\frac{3}{2}+\left(\frac{1}{32}\right)^{2}$$

Square \frac{1}{32} by squaring both the numerator and the denominator of the fraction.

$$p^{2}+\frac{1}{16}p+\frac{1}{1024}=\frac{3}{2}+\frac{1}{1024}$$

Add \frac{3}{2} to \frac{1}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}+\frac{1}{16}p+\frac{1}{1024}=\frac{1537}{1024}$$

Factor p^{2}+\frac{1}{16}p+\frac{1}{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}.

$$\left(p+\frac{1}{32}\right)^{2}=\frac{1537}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+\frac{1}{32}\right)^{2}}=\sqrt{\frac{1537}{1024}}$$

Simplify.

$$p+\frac{1}{32}=\frac{\sqrt{1537}}{32} p+\frac{1}{32}=-\frac{\sqrt{1537}}{32}$$

Subtract \frac{1}{32} from both sides of the equation.

$$p=\frac{\sqrt{1537}-1}{32} p=\frac{-\sqrt{1537}-1}{32}$$