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7x^{2}+19x-10000=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.
x=\frac{-19±\sqrt{19^{2}-4\times 7\left(-10000\right)}}{2\times 7}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 19 for b, and -10000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-19±\sqrt{361-4\times 7\left(-10000\right)}}{2\times 7}
Square 19.
x=\frac{-19±\sqrt{361-28\left(-10000\right)}}{2\times 7}
Multiply -4 times 7.
x=\frac{-19±\sqrt{361+280000}}{2\times 7}
Multiply -28 times -10000.
x=\frac{-19±\sqrt{280361}}{2\times 7}
Add 361 to 280000.
x=\frac{-19±\sqrt{280361}}{14}
Multiply 2 times 7.
x=\frac{\sqrt{280361}-19}{14}
Now solve the equation x=\frac{-19±\sqrt{280361}}{14} when ± is plus. Add -19 to \sqrt{280361}.
x=\frac{-\sqrt{280361}-19}{14}
Now solve the equation x=\frac{-19±\sqrt{280361}}{14} when ± is minus. Subtract \sqrt{280361} from -19.
x=\frac{\sqrt{280361}-19}{14} x=\frac{-\sqrt{280361}-19}{14}
The equation is now solved.
7x^{2}+19x-10000=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.
7x^{2}+19x-10000-\left(-10000\right)=-\left(-10000\right)
Add 10000 to both sides of the equation.
7x^{2}+19x=-\left(-10000\right)
Subtracting -10000 from itself leaves 0.
7x^{2}+19x=10000
Subtract -10000 from 0.
\frac{7x^{2}+19x}{7}=\frac{10000}{7}
Divide both sides by 7.
x^{2}+\frac{19}{7}x=\frac{10000}{7}
Dividing by 7 undoes the multiplication by 7.
x^{2}+\frac{19}{7}x+\left(\frac{19}{14}\right)^{2}=\frac{10000}{7}+\left(\frac{19}{14}\right)^{2}
Divide \frac{19}{7}, the coefficient of the x term, by 2 to get \frac{19}{14}. Then add the square of \frac{19}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{19}{7}x+\frac{361}{196}=\frac{10000}{7}+\frac{361}{196}
Square \frac{19}{14} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{19}{7}x+\frac{361}{196}=\frac{280361}{196}
Add \frac{10000}{7} to \frac{361}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{19}{14}\right)^{2}=\frac{280361}{196}
Factor x^{2}+\frac{19}{7}x+\frac{361}{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}.
\sqrt{\left(x+\frac{19}{14}\right)^{2}}=\sqrt{\frac{280361}{196}}
Take the square root of both sides of the equation.
x+\frac{19}{14}=\frac{\sqrt{280361}}{14} x+\frac{19}{14}=-\frac{\sqrt{280361}}{14}
Simplify.
x=\frac{\sqrt{280361}-19}{14} x=\frac{-\sqrt{280361}-19}{14}
Subtract \frac{19}{14} from both sides of the equation.