Solve -(2k+3)/k^2=-1

$$\frac { - ( 2 k + 3 ) } { k ^ { 2 } } = - 1$$

Solve for k

$$k=-1$$

$$k=3$$

Steps Using Factoring

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

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

To find the opposite of 2k+3, find the opposite of each term.

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

Add k^{2} to both sides.

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

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

$$k^{2}-2k-3=0$$

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

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

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-3 b=1$$

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

$$\left(k-3\right)\left(k+1\right)$$

To find equation solutions, solve k-3=0 and k+1=0.

$$k=3 k=-1$$

Steps Using Factoring By Grouping

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

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

To find the opposite of 2k+3, find the opposite of each term.

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

Add k^{2} to both sides.

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

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

$$k^{2}-2k-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as k^{2}+ak+bk-3. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-3\right)=-3$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-3 b=1$$

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

$$\left(k^{2}-3k\right)+\left(k-3\right)$$

Factor out k in k^{2}-3k.

$$k\left(k-3\right)+k-3$$

Factor out common term k-3 by using distributive property.

$$\left(k-3\right)\left(k+1\right)$$

To find equation solutions, solve k-3=0 and k+1=0.

$$k=3 k=-1$$

Steps Using the Quadratic Formula

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

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

To find the opposite of 2k+3, find the opposite of each term.

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

Add k^{2} to both sides.

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

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.

$$k^{2}-2k-3=0$$

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

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

Square -2.

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

Multiply -4 times -3.

$$k=\frac{-\left(-2\right)±\sqrt{4+12}}{2}$$

Add 4 to 12.

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

Take the square root of 16.

$$k=\frac{-\left(-2\right)±4}{2}$$

The opposite of -2 is 2.

$$k=\frac{2±4}{2}$$

Now solve the equation k=\frac{2±4}{2} when ± is plus. Add 2 to 4.

$$k=\frac{6}{2}$$

Divide 6 by 2.

$$k=3$$

Now solve the equation k=\frac{2±4}{2} when ± is minus. Subtract 4 from 2.

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

Divide -2 by 2.

$$k=-1$$

The equation is now solved.

$$k=3 k=-1$$

Steps for Completing the Square

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

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

To find the opposite of 2k+3, find the opposite of each term.

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

Add k^{2} to both sides.

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

Add 3 to both sides. Anything plus zero gives itself.

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

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.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-2k+1=3+1$$

Add 3 to 1.

$$k^{2}-2k+1=4$$

Factor k^{2}-2k+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k-1\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-1\right)^{2}}=\sqrt{4}$$

Simplify.

$$k-1=2 k-1=-2$$

Add 1 to both sides of the equation.

$$k=3 k=-1$$