Solve k^2-31-2k=-6-3k^2-2k

$$k ^ { 2 } - 31 - 2 k = - 6 - 3 k ^ { 2 } - 2 k$$

Solve for k

$$k = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$k = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using Difference of Squares

Subtract -6 from both sides.

$$k^{2}-31-2k-\left(-6\right)=-3k^{2}-2k$$

The opposite of -6 is 6.

$$k^{2}-31-2k+6=-3k^{2}-2k$$

Add 3k^{2} to both sides.

$$k^{2}-31-2k+6+3k^{2}=-2k$$

Add -31 and 6 to get -25.

$$k^{2}-25-2k+3k^{2}=-2k$$

Combine k^{2} and 3k^{2} to get 4k^{2}.

$$4k^{2}-25-2k=-2k$$

Add 2k to both sides.

$$4k^{2}-25-2k+2k=0$$

Combine -2k and 2k to get 0.

$$4k^{2}-25=0$$

Consider 4k^{2}-25. Rewrite 4k^{2}-25 as \left(2k\right)^{2}-5^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2k-5\right)\left(2k+5\right)=0$$

To find equation solutions, solve 2k-5=0 and 2k+5=0.

$$k=\frac{5}{2} k=-\frac{5}{2}$$

Steps by Finding Square Root

Add 3k^{2} to both sides.

$$k^{2}-31-2k+3k^{2}=-6-2k$$

Combine k^{2} and 3k^{2} to get 4k^{2}.

$$4k^{2}-31-2k=-6-2k$$

Add 2k to both sides.

$$4k^{2}-31-2k+2k=-6$$

Combine -2k and 2k to get 0.

$$4k^{2}-31=-6$$

Add 31 to both sides.

$$4k^{2}=-6+31$$

Add -6 and 31 to get 25.

$$4k^{2}=25$$

Divide both sides by 4.

$$k^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$k=\frac{5}{2} k=-\frac{5}{2}$$

Steps Using the Quadratic Formula

Subtract -6 from both sides.

$$k^{2}-31-2k-\left(-6\right)=-3k^{2}-2k$$

The opposite of -6 is 6.

$$k^{2}-31-2k+6=-3k^{2}-2k$$

Add 3k^{2} to both sides.

$$k^{2}-31-2k+6+3k^{2}=-2k$$

Add -31 and 6 to get -25.

$$k^{2}-25-2k+3k^{2}=-2k$$

Combine k^{2} and 3k^{2} to get 4k^{2}.

$$4k^{2}-25-2k=-2k$$

Add 2k to both sides.

$$4k^{2}-25-2k+2k=0$$

Combine -2k and 2k to get 0.

$$4k^{2}-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 0 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{0±\sqrt{0^{2}-4\times 4\left(-25\right)}}{2\times 4}$$

Square 0.

$$k=\frac{0±\sqrt{-4\times 4\left(-25\right)}}{2\times 4}$$

Multiply -4 times 4.

$$k=\frac{0±\sqrt{-16\left(-25\right)}}{2\times 4}$$

Multiply -16 times -25.

$$k=\frac{0±\sqrt{400}}{2\times 4}$$

Take the square root of 400.

$$k=\frac{0±20}{2\times 4}$$

Multiply 2 times 4.

$$k=\frac{0±20}{8}$$

Now solve the equation k=\frac{0±20}{8} when ± is plus. Reduce the fraction \frac{20}{8} to lowest terms by extracting and canceling out 4.

$$k=\frac{5}{2}$$

Now solve the equation k=\frac{0±20}{8} when ± is minus. Reduce the fraction \frac{-20}{8} to lowest terms by extracting and canceling out 4.

$$k=-\frac{5}{2}$$

The equation is now solved.

$$k=\frac{5}{2} k=-\frac{5}{2}$$