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3x^{2}+10x-5=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{-10±\sqrt{10^{2}-4\times 3\left(-5\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 10 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-10±\sqrt{100-4\times 3\left(-5\right)}}{2\times 3}
Square 10.
x=\frac{-10±\sqrt{100-12\left(-5\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-10±\sqrt{100+60}}{2\times 3}
Multiply -12 times -5.
x=\frac{-10±\sqrt{160}}{2\times 3}
Add 100 to 60.
x=\frac{-10±4\sqrt{10}}{2\times 3}
Take the square root of 160.
x=\frac{-10±4\sqrt{10}}{6}
Multiply 2 times 3.
x=\frac{4\sqrt{10}-10}{6}
Now solve the equation x=\frac{-10±4\sqrt{10}}{6} when ± is plus. Add -10 to 4\sqrt{10}.
x=\frac{2\sqrt{10}-5}{3}
Divide -10+4\sqrt{10} by 6.
x=\frac{-4\sqrt{10}-10}{6}
Now solve the equation x=\frac{-10±4\sqrt{10}}{6} when ± is minus. Subtract 4\sqrt{10} from -10.
x=\frac{-2\sqrt{10}-5}{3}
Divide -10-4\sqrt{10} by 6.
x=\frac{2\sqrt{10}-5}{3} x=\frac{-2\sqrt{10}-5}{3}
The equation is now solved.
3x^{2}+10x-5=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}+10x-5-\left(-5\right)=-\left(-5\right)
Add 5 to both sides of the equation.
3x^{2}+10x=-\left(-5\right)
Subtracting -5 from itself leaves 0.
3x^{2}+10x=5
Subtract -5 from 0.
\frac{3x^{2}+10x}{3}=\frac{5}{3}
Divide both sides by 3.
x^{2}+\frac{10}{3}x=\frac{5}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{10}{3}x+\left(\frac{5}{3}\right)^{2}=\frac{5}{3}+\left(\frac{5}{3}\right)^{2}
Divide \frac{10}{3}, the coefficient of the x term, by 2 to get \frac{5}{3}. Then add the square of \frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{10}{3}x+\frac{25}{9}=\frac{5}{3}+\frac{25}{9}
Square \frac{5}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{10}{3}x+\frac{25}{9}=\frac{40}{9}
Add \frac{5}{3} to \frac{25}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{5}{3}\right)^{2}=\frac{40}{9}
Factor x^{2}+\frac{10}{3}x+\frac{25}{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{5}{3}\right)^{2}}=\sqrt{\frac{40}{9}}
Take the square root of both sides of the equation.
x+\frac{5}{3}=\frac{2\sqrt{10}}{3} x+\frac{5}{3}=-\frac{2\sqrt{10}}{3}
Simplify.
x=\frac{2\sqrt{10}-5}{3} x=\frac{-2\sqrt{10}-5}{3}
Subtract \frac{5}{3} from both sides of the equation.