Solve 4z^2+60z=800

$$4 z ^ { 2 } + 60 z = 800$$

Solve for z

$$z = \frac{5 \sqrt{41} - 15}{2} \approx 8.507810594$$

$$z=\frac{-5\sqrt{41}-15}{2}\approx -23.507810594$$

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.

$$4z^{2}+60z=800$$

Subtract 800 from both sides of the equation.

$$4z^{2}+60z-800=800-800$$

Subtracting 800 from itself leaves 0.

$$4z^{2}+60z-800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 60 for b, and -800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-60±\sqrt{60^{2}-4\times 4\left(-800\right)}}{2\times 4}$$

Square 60.

$$z=\frac{-60±\sqrt{3600-4\times 4\left(-800\right)}}{2\times 4}$$

Multiply -4 times 4.

$$z=\frac{-60±\sqrt{3600-16\left(-800\right)}}{2\times 4}$$

Multiply -16 times -800.

$$z=\frac{-60±\sqrt{3600+12800}}{2\times 4}$$

Add 3600 to 12800.

$$z=\frac{-60±\sqrt{16400}}{2\times 4}$$

Take the square root of 16400.

$$z=\frac{-60±20\sqrt{41}}{2\times 4}$$

Multiply 2 times 4.

$$z=\frac{-60±20\sqrt{41}}{8}$$

Now solve the equation z=\frac{-60±20\sqrt{41}}{8} when ± is plus. Add -60 to 20\sqrt{41}.

$$z=\frac{20\sqrt{41}-60}{8}$$

Divide -60+20\sqrt{41} by 8.

$$z=\frac{5\sqrt{41}-15}{2}$$

Now solve the equation z=\frac{-60±20\sqrt{41}}{8} when ± is minus. Subtract 20\sqrt{41} from -60.

$$z=\frac{-20\sqrt{41}-60}{8}$$

Divide -60-20\sqrt{41} by 8.

$$z=\frac{-5\sqrt{41}-15}{2}$$

The equation is now solved.

$$z=\frac{5\sqrt{41}-15}{2} z=\frac{-5\sqrt{41}-15}{2}$$

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.

$$4z^{2}+60z=800$$

Divide both sides by 4.

$$\frac{4z^{2}+60z}{4}=\frac{800}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$z^{2}+\frac{60}{4}z=\frac{800}{4}$$

Divide 60 by 4.

$$z^{2}+15z=\frac{800}{4}$$

Divide 800 by 4.

$$z^{2}+15z=200$$

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

$$z^{2}+15z+\left(\frac{15}{2}\right)^{2}=200+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$z^{2}+15z+\frac{225}{4}=200+\frac{225}{4}$$

Add 200 to \frac{225}{4}.

$$z^{2}+15z+\frac{225}{4}=\frac{1025}{4}$$

Factor z^{2}+15z+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z+\frac{15}{2}\right)^{2}=\frac{1025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+\frac{15}{2}\right)^{2}}=\sqrt{\frac{1025}{4}}$$

Simplify.

$$z+\frac{15}{2}=\frac{5\sqrt{41}}{2} z+\frac{15}{2}=-\frac{5\sqrt{41}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$z=\frac{5\sqrt{41}-15}{2} z=\frac{-5\sqrt{41}-15}{2}$$