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4z^{2}+60z=800
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=800-800
Subtract 800 from both sides of the equation.
4z^{2}+60z-800=0
Subtracting 800 from itself leaves 0.
z=\frac{-60±\sqrt{60^{2}-4\times 4\left(-800\right)}}{2\times 4}
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{3600-4\times 4\left(-800\right)}}{2\times 4}
Square 60.
z=\frac{-60±\sqrt{3600-16\left(-800\right)}}{2\times 4}
Multiply -4 times 4.
z=\frac{-60±\sqrt{3600+12800}}{2\times 4}
Multiply -16 times -800.
z=\frac{-60±\sqrt{16400}}{2\times 4}
Add 3600 to 12800.
z=\frac{-60±20\sqrt{41}}{2\times 4}
Take the square root of 16400.
z=\frac{-60±20\sqrt{41}}{8}
Multiply 2 times 4.
z=\frac{20\sqrt{41}-60}{8}
Now solve the equation z=\frac{-60±20\sqrt{41}}{8} when ± is plus. Add -60 to 20\sqrt{41}.
z=\frac{5\sqrt{41}-15}{2}
Divide -60+20\sqrt{41} by 8.
z=\frac{-20\sqrt{41}-60}{8}
Now solve the equation z=\frac{-60±20\sqrt{41}}{8} when ± is minus. Subtract 20\sqrt{41} from -60.
z=\frac{-5\sqrt{41}-15}{2}
Divide -60-20\sqrt{41} by 8.
z=\frac{5\sqrt{41}-15}{2} z=\frac{-5\sqrt{41}-15}{2}
The equation is now solved.
4z^{2}+60z=800
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.
\frac{4z^{2}+60z}{4}=\frac{800}{4}
Divide both sides by 4.
z^{2}+\frac{60}{4}z=\frac{800}{4}
Dividing by 4 undoes the multiplication by 4.
z^{2}+15z=\frac{800}{4}
Divide 60 by 4.
z^{2}+15z=200
Divide 800 by 4.
z^{2}+15z+\left(\frac{15}{2}\right)^{2}=200+\left(\frac{15}{2}\right)^{2}
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+\frac{225}{4}=200+\frac{225}{4}
Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.
z^{2}+15z+\frac{225}{4}=\frac{1025}{4}
Add 200 to \frac{225}{4}.
\left(z+\frac{15}{2}\right)^{2}=\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}.
\sqrt{\left(z+\frac{15}{2}\right)^{2}}=\sqrt{\frac{1025}{4}}
Take the square root of both sides of the equation.
z+\frac{15}{2}=\frac{5\sqrt{41}}{2} z+\frac{15}{2}=-\frac{5\sqrt{41}}{2}
Simplify.
z=\frac{5\sqrt{41}-15}{2} z=\frac{-5\sqrt{41}-15}{2}
Subtract \frac{15}{2} from both sides of the equation.