Solve 2w^2-11w=-12

$$2 w ^ { 2 } - 11 w = - 12$$

Solve for w

$$w = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$w=4$$

Steps Using Factoring By Grouping

Add 12 to both sides.

$$2w^{2}-11w+12=0$$

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

$$a+b=-11 ab=2\times 12=24$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -11.

$$a=-8 b=-3$$

Rewrite 2w^{2}-11w+12 as \left(2w^{2}-8w\right)+\left(-3w+12\right).

$$\left(2w^{2}-8w\right)+\left(-3w+12\right)$$

Factor out 2w in the first and -3 in the second group.

$$2w\left(w-4\right)-3\left(w-4\right)$$

Factor out common term w-4 by using distributive property.

$$\left(w-4\right)\left(2w-3\right)$$

To find equation solutions, solve w-4=0 and 2w-3=0.

$$w=4 w=\frac{3}{2}$$

Steps Using the Quadratic Formula

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.

$$2w^{2}-11w=-12$$

Add 12 to both sides of the equation.

$$2w^{2}-11w-\left(-12\right)=-12-\left(-12\right)$$

Subtracting -12 from itself leaves 0.

$$2w^{2}-11w-\left(-12\right)=0$$

Subtract -12 from 0.

$$2w^{2}-11w+12=0$$

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

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

Square -11.

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

Multiply -4 times 2.

$$w=\frac{-\left(-11\right)±\sqrt{121-8\times 12}}{2\times 2}$$

Multiply -8 times 12.

$$w=\frac{-\left(-11\right)±\sqrt{121-96}}{2\times 2}$$

Add 121 to -96.

$$w=\frac{-\left(-11\right)±\sqrt{25}}{2\times 2}$$

Take the square root of 25.

$$w=\frac{-\left(-11\right)±5}{2\times 2}$$

The opposite of -11 is 11.

$$w=\frac{11±5}{2\times 2}$$

Multiply 2 times 2.

$$w=\frac{11±5}{4}$$

Now solve the equation w=\frac{11±5}{4} when ± is plus. Add 11 to 5.

$$w=\frac{16}{4}$$

Divide 16 by 4.

$$w=4$$

Now solve the equation w=\frac{11±5}{4} when ± is minus. Subtract 5 from 11.

$$w=\frac{6}{4}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$w=4 w=\frac{3}{2}$$

Steps for Completing the Square

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.

$$2w^{2}-11w=-12$$

Divide both sides by 2.

$$\frac{2w^{2}-11w}{2}=-\frac{12}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$w^{2}-\frac{11}{2}w=-\frac{12}{2}$$

Divide -12 by 2.

$$w^{2}-\frac{11}{2}w=-6$$

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

$$w^{2}-\frac{11}{2}w+\left(-\frac{11}{4}\right)^{2}=-6+\left(-\frac{11}{4}\right)^{2}$$

Square -\frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$w^{2}-\frac{11}{2}w+\frac{121}{16}=-6+\frac{121}{16}$$

Add -6 to \frac{121}{16}.

$$w^{2}-\frac{11}{2}w+\frac{121}{16}=\frac{25}{16}$$

Factor w^{2}-\frac{11}{2}w+\frac{121}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w-\frac{11}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-\frac{11}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$w-\frac{11}{4}=\frac{5}{4} w-\frac{11}{4}=-\frac{5}{4}$$

Add \frac{11}{4} to both sides of the equation.

$$w=4 w=\frac{3}{2}$$