Solve 2/w^2-3/w+1=0

$$\frac { 2 } { w ^ { 2 } } - \frac { 3 } { w } + 1 = 0$$

Solve for w

$$w=1$$

$$w=2$$

Steps Using Factoring

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w^{2}, the least common multiple of w^{2},w.

$$2-w\times 3+w^{2}=0$$

Multiply -1 and 3 to get -3.

$$2-3w+w^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$w^{2}-3w+2=0$$

To solve the equation, factor w^{2}-3w+2 using formula w^{2}+\left(a+b\right)w+ab=\left(w+a\right)\left(w+b\right). To find a and b, set up a system to be solved.

$$a+b=-3 ab=2$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-2 b=-1$$

Rewrite factored expression \left(w+a\right)\left(w+b\right) using the obtained values.

$$\left(w-2\right)\left(w-1\right)$$

To find equation solutions, solve w-2=0 and w-1=0.

$$w=2 w=1$$

Steps Using Factoring By Grouping

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w^{2}, the least common multiple of w^{2},w.

$$2-w\times 3+w^{2}=0$$

Multiply -1 and 3 to get -3.

$$2-3w+w^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$w^{2}-3w+2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as w^{2}+aw+bw+2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\times 2=2$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-2 b=-1$$

Rewrite w^{2}-3w+2 as \left(w^{2}-2w\right)+\left(-w+2\right).

$$\left(w^{2}-2w\right)+\left(-w+2\right)$$

Factor out w in the first and -1 in the second group.

$$w\left(w-2\right)-\left(w-2\right)$$

Factor out common term w-2 by using distributive property.

$$\left(w-2\right)\left(w-1\right)$$

To find equation solutions, solve w-2=0 and w-1=0.

$$w=2 w=1$$

Steps Using the Quadratic Formula

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w^{2}, the least common multiple of w^{2},w.

$$2-w\times 3+w^{2}=0$$

Multiply -1 and 3 to get -3.

$$2-3w+w^{2}=0$$

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.

$$w^{2}-3w+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -3.

$$w=\frac{-\left(-3\right)±\sqrt{9-4\times 2}}{2}$$

Multiply -4 times 2.

$$w=\frac{-\left(-3\right)±\sqrt{9-8}}{2}$$

Add 9 to -8.

$$w=\frac{-\left(-3\right)±\sqrt{1}}{2}$$

Take the square root of 1.

$$w=\frac{-\left(-3\right)±1}{2}$$

The opposite of -3 is 3.

$$w=\frac{3±1}{2}$$

Now solve the equation w=\frac{3±1}{2} when ± is plus. Add 3 to 1.

$$w=\frac{4}{2}$$

Divide 4 by 2.

$$w=2$$

Now solve the equation w=\frac{3±1}{2} when ± is minus. Subtract 1 from 3.

$$w=\frac{2}{2}$$

Divide 2 by 2.

$$w=1$$

The equation is now solved.

$$w=2 w=1$$

Steps for Completing the Square

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w^{2}, the least common multiple of w^{2},w.

$$2-w\times 3+w^{2}=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$-w\times 3+w^{2}=-2$$

Multiply -1 and 3 to get -3.

$$-3w+w^{2}=-2$$

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.

$$w^{2}-3w=-2$$

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

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

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

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

Add -2 to \frac{9}{4}.

$$w^{2}-3w+\frac{9}{4}=\frac{1}{4}$$

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

$$\left(w-\frac{3}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{3}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$w-\frac{3}{2}=\frac{1}{2} w-\frac{3}{2}=-\frac{1}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$w=2 w=1$$