Solve -30k=-23k^2

$$- 30 k = - 23 k ^ { 2 }$$

Solve for k

$$k = \frac{30}{23} = 1\frac{7}{23} \approx 1.304347826$$

$$k=0$$

Steps Using Factoring

Add 23k^{2} to both sides.

$$-30k+23k^{2}=0$$

Factor out k.

$$k\left(-30+23k\right)=0$$

To find equation solutions, solve k=0 and -30+23k=0.

$$k=0 k=\frac{30}{23}$$

Steps Using the Quadratic Formula

Add 23k^{2} to both sides.

$$-30k+23k^{2}=0$$

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.

$$23k^{2}-30k=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 23 for a, -30 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}}}{2\times 23}$$

Take the square root of \left(-30\right)^{2}.

$$k=\frac{-\left(-30\right)±30}{2\times 23}$$

The opposite of -30 is 30.

$$k=\frac{30±30}{2\times 23}$$

Multiply 2 times 23.

$$k=\frac{30±30}{46}$$

Now solve the equation k=\frac{30±30}{46} when ± is plus. Add 30 to 30.

$$k=\frac{60}{46}$$

Reduce the fraction \frac{60}{46} to lowest terms by extracting and canceling out 2.

$$k=\frac{30}{23}$$

Now solve the equation k=\frac{30±30}{46} when ± is minus. Subtract 30 from 30.

$$k=\frac{0}{46}$$

Divide 0 by 46.

$$k=0$$

The equation is now solved.

$$k=\frac{30}{23} k=0$$

Steps for Completing the Square

Add 23k^{2} to both sides.

$$-30k+23k^{2}=0$$

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.

$$23k^{2}-30k=0$$

Divide both sides by 23.

$$\frac{23k^{2}-30k}{23}=\frac{0}{23}$$

Dividing by 23 undoes the multiplication by 23.

$$k^{2}-\frac{30}{23}k=\frac{0}{23}$$

Divide 0 by 23.

$$k^{2}-\frac{30}{23}k=0$$

Divide -\frac{30}{23}, the coefficient of the x term, by 2 to get -\frac{15}{23}. Then add the square of -\frac{15}{23} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-\frac{30}{23}k+\left(-\frac{15}{23}\right)^{2}=\left(-\frac{15}{23}\right)^{2}$$

Square -\frac{15}{23} by squaring both the numerator and the denominator of the fraction.

$$k^{2}-\frac{30}{23}k+\frac{225}{529}=\frac{225}{529}$$

Factor k^{2}-\frac{30}{23}k+\frac{225}{529}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k-\frac{15}{23}\right)^{2}=\frac{225}{529}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-\frac{15}{23}\right)^{2}}=\sqrt{\frac{225}{529}}$$

Simplify.

$$k-\frac{15}{23}=\frac{15}{23} k-\frac{15}{23}=-\frac{15}{23}$$

Add \frac{15}{23} to both sides of the equation.

$$k=\frac{30}{23} k=0$$