Solve p^-2+11p^-1-12=0

$$p ^ { - 2 } + 11 p ^ { - 1 } - 12 = 0$$

Solve for p

$$p=1$$

$$p=-\frac{1}{12}\approx -0.083333333$$

Steps Using Factoring By Grouping

Reorder the terms.

$$-12+11\times \frac{1}{p}+p^{-2}=0$$

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

$$p\left(-12\right)+11\times 1+pp^{-2}=0$$

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

$$p\left(-12\right)+11\times 1+p^{-1}=0$$

Multiply 11 and 1 to get 11.

$$p\left(-12\right)+11+p^{-1}=0$$

Reorder the terms.

$$-12p+11+\frac{1}{p}=0$$

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

$$-12pp+p\times 11+1=0$$

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

$$-12p^{2}+p\times 11+1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -12p^{2}+ap+bp+1. To find a and b, set up a system to be solved.

$$a+b=11 ab=-12=-12$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 11.

$$a=12 b=-1$$

Rewrite -12p^{2}+11p+1 as \left(-12p^{2}+12p\right)+\left(-p+1\right).

$$\left(-12p^{2}+12p\right)+\left(-p+1\right)$$

Factor out 12p in -12p^{2}+12p.

$$12p\left(-p+1\right)-p+1$$

Factor out common term -p+1 by using distributive property.

$$\left(-p+1\right)\left(12p+1\right)$$

To find equation solutions, solve -p+1=0 and 12p+1=0.

$$p=1 p=-\frac{1}{12}$$

Steps Using the Quadratic Formula

Reorder the terms.

$$-12+11\times \frac{1}{p}+p^{-2}=0$$

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

$$p\left(-12\right)+11\times 1+pp^{-2}=0$$

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

$$p\left(-12\right)+11\times 1+p^{-1}=0$$

Multiply 11 and 1 to get 11.

$$p\left(-12\right)+11+p^{-1}=0$$

Reorder the terms.

$$-12p+11+\frac{1}{p}=0$$

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

$$-12pp+p\times 11+1=0$$

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

$$-12p^{2}+p\times 11+1=0$$

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.

$$-12p^{2}+11p+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, 11 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-11±\sqrt{11^{2}-4\left(-12\right)}}{2\left(-12\right)}$$

Square 11.

$$p=\frac{-11±\sqrt{121-4\left(-12\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$p=\frac{-11±\sqrt{121+48}}{2\left(-12\right)}$$

Add 121 to 48.

$$p=\frac{-11±\sqrt{169}}{2\left(-12\right)}$$

Take the square root of 169.

$$p=\frac{-11±13}{2\left(-12\right)}$$

Multiply 2 times -12.

$$p=\frac{-11±13}{-24}$$

Now solve the equation p=\frac{-11±13}{-24} when ± is plus. Add -11 to 13.

$$p=\frac{2}{-24}$$

Reduce the fraction \frac{2}{-24} to lowest terms by extracting and canceling out 2.

$$p=-\frac{1}{12}$$

Now solve the equation p=\frac{-11±13}{-24} when ± is minus. Subtract 13 from -11.

$$p=-\frac{24}{-24}$$

Divide -24 by -24.

$$p=1$$

The equation is now solved.

$$p=-\frac{1}{12} p=1$$

Steps for Completing the Square

Add 12 to both sides. Anything plus zero gives itself.

$$p^{-2}+11p^{-1}=12$$

Reorder the terms.

$$11\times \frac{1}{p}+p^{-2}=12$$

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

$$11\times 1+pp^{-2}=12p$$

To multiply powers of the same base, add their exponents. Add 1 and -2 to get -1.

$$11\times 1+p^{-1}=12p$$

Multiply 11 and 1 to get 11.

$$11+p^{-1}=12p$$

Subtract 12p from both sides.

$$11+p^{-1}-12p=0$$

Subtract 11 from both sides. Anything subtracted from zero gives its negation.

$$p^{-1}-12p=-11$$

Reorder the terms.

$$-12p+\frac{1}{p}=-11$$

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

$$-12pp+1=-11p$$

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

$$-12p^{2}+1=-11p$$

Add 11p to both sides.

$$-12p^{2}+1+11p=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-12p^{2}+11p=-1$$

Divide both sides by -12.

$$\frac{-12p^{2}+11p}{-12}=-\frac{1}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$p^{2}+\frac{11}{-12}p=-\frac{1}{-12}$$

Divide 11 by -12.

$$p^{2}-\frac{11}{12}p=-\frac{1}{-12}$$

Divide -1 by -12.

$$p^{2}-\frac{11}{12}p=\frac{1}{12}$$

Divide -\frac{11}{12}, the coefficient of the x term, by 2 to get -\frac{11}{24}. Then add the square of -\frac{11}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{11}{12}p+\left(-\frac{11}{24}\right)^{2}=\frac{1}{12}+\left(-\frac{11}{24}\right)^{2}$$

Square -\frac{11}{24} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-\frac{11}{12}p+\frac{121}{576}=\frac{1}{12}+\frac{121}{576}$$

Add \frac{1}{12} to \frac{121}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}-\frac{11}{12}p+\frac{121}{576}=\frac{169}{576}$$

Factor p^{2}-\frac{11}{12}p+\frac{121}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-\frac{11}{24}\right)^{2}=\frac{169}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{11}{24}\right)^{2}}=\sqrt{\frac{169}{576}}$$

Simplify.

$$p-\frac{11}{24}=\frac{13}{24} p-\frac{11}{24}=-\frac{13}{24}$$

Add \frac{11}{24} to both sides of the equation.

$$p=1 p=-\frac{1}{12}$$