Solve 4f^2-112f=-52

$$4 f ^ { 2 } - 112 f = - 52$$

Solve for f

$$f=\sqrt{183}+14\approx 27.527749258$$

$$f=14-\sqrt{183}\approx 0.472250742$$

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.

$$4f^{2}-112f=-52$$

Add 52 to both sides of the equation.

$$4f^{2}-112f-\left(-52\right)=-52-\left(-52\right)$$

Subtracting -52 from itself leaves 0.

$$4f^{2}-112f-\left(-52\right)=0$$

Subtract -52 from 0.

$$4f^{2}-112f+52=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -112 for b, and 52 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$f=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\times 4\times 52}}{2\times 4}$$

Square -112.

$$f=\frac{-\left(-112\right)±\sqrt{12544-4\times 4\times 52}}{2\times 4}$$

Multiply -4 times 4.

$$f=\frac{-\left(-112\right)±\sqrt{12544-16\times 52}}{2\times 4}$$

Multiply -16 times 52.

$$f=\frac{-\left(-112\right)±\sqrt{12544-832}}{2\times 4}$$

Add 12544 to -832.

$$f=\frac{-\left(-112\right)±\sqrt{11712}}{2\times 4}$$

Take the square root of 11712.

$$f=\frac{-\left(-112\right)±8\sqrt{183}}{2\times 4}$$

The opposite of -112 is 112.

$$f=\frac{112±8\sqrt{183}}{2\times 4}$$

Multiply 2 times 4.

$$f=\frac{112±8\sqrt{183}}{8}$$

Now solve the equation f=\frac{112±8\sqrt{183}}{8} when ± is plus. Add 112 to 8\sqrt{183}.

$$f=\frac{8\sqrt{183}+112}{8}$$

Divide 112+8\sqrt{183} by 8.

$$f=\sqrt{183}+14$$

Now solve the equation f=\frac{112±8\sqrt{183}}{8} when ± is minus. Subtract 8\sqrt{183} from 112.

$$f=\frac{112-8\sqrt{183}}{8}$$

Divide 112-8\sqrt{183} by 8.

$$f=14-\sqrt{183}$$

The equation is now solved.

$$f=\sqrt{183}+14 f=14-\sqrt{183}$$

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.

$$4f^{2}-112f=-52$$

Divide both sides by 4.

$$\frac{4f^{2}-112f}{4}=-\frac{52}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$f^{2}+\left(-\frac{112}{4}\right)f=-\frac{52}{4}$$

Divide -112 by 4.

$$f^{2}-28f=-\frac{52}{4}$$

Divide -52 by 4.

$$f^{2}-28f=-13$$

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$f^{2}-28f+\left(-14\right)^{2}=-13+\left(-14\right)^{2}$$

Square -14.

$$f^{2}-28f+196=-13+196$$

Add -13 to 196.

$$f^{2}-28f+196=183$$

Factor f^{2}-28f+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(f-14\right)^{2}=183$$

Take the square root of both sides of the equation.

$$\sqrt{\left(f-14\right)^{2}}=\sqrt{183}$$

Simplify.

$$f-14=\sqrt{183} f-14=-\sqrt{183}$$

Add 14 to both sides of the equation.

$$f=\sqrt{183}+14 f=14-\sqrt{183}$$