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Solve for x (complex solution)
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10x^{2}+60x+30=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{-60±\sqrt{60^{2}-4\times 10\times 30}}{2\times 10}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 60 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-60±\sqrt{3600-4\times 10\times 30}}{2\times 10}
Square 60.
x=\frac{-60±\sqrt{3600-40\times 30}}{2\times 10}
Multiply -4 times 10.
x=\frac{-60±\sqrt{3600-1200}}{2\times 10}
Multiply -40 times 30.
x=\frac{-60±\sqrt{2400}}{2\times 10}
Add 3600 to -1200.
x=\frac{-60±20\sqrt{6}}{2\times 10}
Take the square root of 2400.
x=\frac{-60±20\sqrt{6}}{20}
Multiply 2 times 10.
x=\frac{20\sqrt{6}-60}{20}
Now solve the equation x=\frac{-60±20\sqrt{6}}{20} when ± is plus. Add -60 to 20\sqrt{6}.
x=\sqrt{6}-3
Divide -60+20\sqrt{6} by 20.
x=\frac{-20\sqrt{6}-60}{20}
Now solve the equation x=\frac{-60±20\sqrt{6}}{20} when ± is minus. Subtract 20\sqrt{6} from -60.
x=-\sqrt{6}-3
Divide -60-20\sqrt{6} by 20.
x=\sqrt{6}-3 x=-\sqrt{6}-3
The equation is now solved.
10x^{2}+60x+30=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.
10x^{2}+60x+30-30=-30
Subtract 30 from both sides of the equation.
10x^{2}+60x=-30
Subtracting 30 from itself leaves 0.
\frac{10x^{2}+60x}{10}=-\frac{30}{10}
Divide both sides by 10.
x^{2}+\frac{60}{10}x=-\frac{30}{10}
Dividing by 10 undoes the multiplication by 10.
x^{2}+6x=-\frac{30}{10}
Divide 60 by 10.
x^{2}+6x=-3
Divide -30 by 10.
x^{2}+6x+3^{2}=-3+3^{2}
Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+6x+9=-3+9
Square 3.
x^{2}+6x+9=6
Add -3 to 9.
\left(x+3\right)^{2}=6
Factor x^{2}+6x+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+3\right)^{2}}=\sqrt{6}
Take the square root of both sides of the equation.
x+3=\sqrt{6} x+3=-\sqrt{6}
Simplify.
x=\sqrt{6}-3 x=-\sqrt{6}-3
Subtract 3 from both sides of the equation.
10x^{2}+60x+30=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{-60±\sqrt{60^{2}-4\times 10\times 30}}{2\times 10}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 60 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-60±\sqrt{3600-4\times 10\times 30}}{2\times 10}
Square 60.
x=\frac{-60±\sqrt{3600-40\times 30}}{2\times 10}
Multiply -4 times 10.
x=\frac{-60±\sqrt{3600-1200}}{2\times 10}
Multiply -40 times 30.
x=\frac{-60±\sqrt{2400}}{2\times 10}
Add 3600 to -1200.
x=\frac{-60±20\sqrt{6}}{2\times 10}
Take the square root of 2400.
x=\frac{-60±20\sqrt{6}}{20}
Multiply 2 times 10.
x=\frac{20\sqrt{6}-60}{20}
Now solve the equation x=\frac{-60±20\sqrt{6}}{20} when ± is plus. Add -60 to 20\sqrt{6}.
x=\sqrt{6}-3
Divide -60+20\sqrt{6} by 20.
x=\frac{-20\sqrt{6}-60}{20}
Now solve the equation x=\frac{-60±20\sqrt{6}}{20} when ± is minus. Subtract 20\sqrt{6} from -60.
x=-\sqrt{6}-3
Divide -60-20\sqrt{6} by 20.
x=\sqrt{6}-3 x=-\sqrt{6}-3
The equation is now solved.
10x^{2}+60x+30=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.
10x^{2}+60x+30-30=-30
Subtract 30 from both sides of the equation.
10x^{2}+60x=-30
Subtracting 30 from itself leaves 0.
\frac{10x^{2}+60x}{10}=-\frac{30}{10}
Divide both sides by 10.
x^{2}+\frac{60}{10}x=-\frac{30}{10}
Dividing by 10 undoes the multiplication by 10.
x^{2}+6x=-\frac{30}{10}
Divide 60 by 10.
x^{2}+6x=-3
Divide -30 by 10.
x^{2}+6x+3^{2}=-3+3^{2}
Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+6x+9=-3+9
Square 3.
x^{2}+6x+9=6
Add -3 to 9.
\left(x+3\right)^{2}=6
Factor x^{2}+6x+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+3\right)^{2}}=\sqrt{6}
Take the square root of both sides of the equation.
x+3=\sqrt{6} x+3=-\sqrt{6}
Simplify.
x=\sqrt{6}-3 x=-\sqrt{6}-3
Subtract 3 from both sides of the equation.