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3x^{2}-50x-26=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{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 3\left(-26\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -50 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-50\right)±\sqrt{2500-4\times 3\left(-26\right)}}{2\times 3}
Square -50.
x=\frac{-\left(-50\right)±\sqrt{2500-12\left(-26\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-50\right)±\sqrt{2500+312}}{2\times 3}
Multiply -12 times -26.
x=\frac{-\left(-50\right)±\sqrt{2812}}{2\times 3}
Add 2500 to 312.
x=\frac{-\left(-50\right)±2\sqrt{703}}{2\times 3}
Take the square root of 2812.
x=\frac{50±2\sqrt{703}}{2\times 3}
The opposite of -50 is 50.
x=\frac{50±2\sqrt{703}}{6}
Multiply 2 times 3.
x=\frac{2\sqrt{703}+50}{6}
Now solve the equation x=\frac{50±2\sqrt{703}}{6} when ± is plus. Add 50 to 2\sqrt{703}.
x=\frac{\sqrt{703}+25}{3}
Divide 50+2\sqrt{703} by 6.
x=\frac{50-2\sqrt{703}}{6}
Now solve the equation x=\frac{50±2\sqrt{703}}{6} when ± is minus. Subtract 2\sqrt{703} from 50.
x=\frac{25-\sqrt{703}}{3}
Divide 50-2\sqrt{703} by 6.
x=\frac{\sqrt{703}+25}{3} x=\frac{25-\sqrt{703}}{3}
The equation is now solved.
3x^{2}-50x-26=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.
3x^{2}-50x-26-\left(-26\right)=-\left(-26\right)
Add 26 to both sides of the equation.
3x^{2}-50x=-\left(-26\right)
Subtracting -26 from itself leaves 0.
3x^{2}-50x=26
Subtract -26 from 0.
\frac{3x^{2}-50x}{3}=\frac{26}{3}
Divide both sides by 3.
x^{2}-\frac{50}{3}x=\frac{26}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}-\frac{50}{3}x+\left(-\frac{25}{3}\right)^{2}=\frac{26}{3}+\left(-\frac{25}{3}\right)^{2}
Divide -\frac{50}{3}, the coefficient of the x term, by 2 to get -\frac{25}{3}. Then add the square of -\frac{25}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{50}{3}x+\frac{625}{9}=\frac{26}{3}+\frac{625}{9}
Square -\frac{25}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{50}{3}x+\frac{625}{9}=\frac{703}{9}
Add \frac{26}{3} to \frac{625}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{25}{3}\right)^{2}=\frac{703}{9}
Factor x^{2}-\frac{50}{3}x+\frac{625}{9}. 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{25}{3}\right)^{2}}=\sqrt{\frac{703}{9}}
Take the square root of both sides of the equation.
x-\frac{25}{3}=\frac{\sqrt{703}}{3} x-\frac{25}{3}=-\frac{\sqrt{703}}{3}
Simplify.
x=\frac{\sqrt{703}+25}{3} x=\frac{25-\sqrt{703}}{3}
Add \frac{25}{3} to both sides of the equation.