Solve 42f^2-11f=0

$$42 f ^ { 2 } - 11 f = 0$$

Solve for f

$$f=\frac{11}{42}\approx 0.261904762$$

$$f=0$$

Steps Using Factoring

Factor out f.

$$f\left(42f-11\right)=0$$

To find equation solutions, solve f=0 and 42f-11=0.

$$f=0 f=\frac{11}{42}$$

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.

$$42f^{2}-11f=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 42 for a, -11 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$f=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}}}{2\times 42}$$

Take the square root of \left(-11\right)^{2}.

$$f=\frac{-\left(-11\right)±11}{2\times 42}$$

The opposite of -11 is 11.

$$f=\frac{11±11}{2\times 42}$$

Multiply 2 times 42.

$$f=\frac{11±11}{84}$$

Now solve the equation f=\frac{11±11}{84} when ± is plus. Add 11 to 11.

$$f=\frac{22}{84}$$

Reduce the fraction \frac{22}{84} to lowest terms by extracting and canceling out 2.

$$f=\frac{11}{42}$$

Now solve the equation f=\frac{11±11}{84} when ± is minus. Subtract 11 from 11.

$$f=\frac{0}{84}$$

Divide 0 by 84.

$$f=0$$

The equation is now solved.

$$f=\frac{11}{42} f=0$$

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.

$$42f^{2}-11f=0$$

Divide both sides by 42.

$$\frac{42f^{2}-11f}{42}=\frac{0}{42}$$

Dividing by 42 undoes the multiplication by 42.

$$f^{2}-\frac{11}{42}f=\frac{0}{42}$$

Divide 0 by 42.

$$f^{2}-\frac{11}{42}f=0$$

Divide -\frac{11}{42}, the coefficient of the x term, by 2 to get -\frac{11}{84}. Then add the square of -\frac{11}{84} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$f^{2}-\frac{11}{42}f+\left(-\frac{11}{84}\right)^{2}=\left(-\frac{11}{84}\right)^{2}$$

Square -\frac{11}{84} by squaring both the numerator and the denominator of the fraction.

$$f^{2}-\frac{11}{42}f+\frac{121}{7056}=\frac{121}{7056}$$

Factor f^{2}-\frac{11}{42}f+\frac{121}{7056}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(f-\frac{11}{84}\right)^{2}=\frac{121}{7056}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(f-\frac{11}{84}\right)^{2}}=\sqrt{\frac{121}{7056}}$$

Simplify.

$$f-\frac{11}{84}=\frac{11}{84} f-\frac{11}{84}=-\frac{11}{84}$$

Add \frac{11}{84} to both sides of the equation.

$$f=\frac{11}{42} f=0$$