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Solve for x (complex solution)
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-5x^{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{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-5\right)\times 5}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -10 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-10\right)±\sqrt{100-4\left(-5\right)\times 5}}{2\left(-5\right)}
Square -10.
x=\frac{-\left(-10\right)±\sqrt{100+20\times 5}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-\left(-10\right)±\sqrt{100+100}}{2\left(-5\right)}
Multiply 20 times 5.
x=\frac{-\left(-10\right)±\sqrt{200}}{2\left(-5\right)}
Add 100 to 100.
x=\frac{-\left(-10\right)±10\sqrt{2}}{2\left(-5\right)}
Take the square root of 200.
x=\frac{10±10\sqrt{2}}{2\left(-5\right)}
The opposite of -10 is 10.
x=\frac{10±10\sqrt{2}}{-10}
Multiply 2 times -5.
x=\frac{10\sqrt{2}+10}{-10}
Now solve the equation x=\frac{10±10\sqrt{2}}{-10} when ± is plus. Add 10 to 10\sqrt{2}.
x=-\left(\sqrt{2}+1\right)
Divide 10+10\sqrt{2} by -10.
x=\frac{10-10\sqrt{2}}{-10}
Now solve the equation x=\frac{10±10\sqrt{2}}{-10} when ± is minus. Subtract 10\sqrt{2} from 10.
x=\sqrt{2}-1
Divide 10-10\sqrt{2} by -10.
x=-\left(\sqrt{2}+1\right) x=\sqrt{2}-1
The equation is now solved.
-5x^{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.
-5x^{2}-10x+5-5=-5
Subtract 5 from both sides of the equation.
-5x^{2}-10x=-5
Subtracting 5 from itself leaves 0.
\frac{-5x^{2}-10x}{-5}=-\frac{5}{-5}
Divide both sides by -5.
x^{2}+\left(-\frac{10}{-5}\right)x=-\frac{5}{-5}
Dividing by -5 undoes the multiplication by -5.
x^{2}+2x=-\frac{5}{-5}
Divide -10 by -5.
x^{2}+2x=1
Divide -5 by -5.
x^{2}+2x+1^{2}=1+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=1+1
Square 1.
x^{2}+2x+1=2
Add 1 to 1.
\left(x+1\right)^{2}=2
Factor x^{2}+2x+1. 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+1\right)^{2}}=\sqrt{2}
Take the square root of both sides of the equation.
x+1=\sqrt{2} x+1=-\sqrt{2}
Simplify.
x=\sqrt{2}-1 x=-\sqrt{2}-1
Subtract 1 from both sides of the equation.
-5x^{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{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-5\right)\times 5}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -10 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-10\right)±\sqrt{100-4\left(-5\right)\times 5}}{2\left(-5\right)}
Square -10.
x=\frac{-\left(-10\right)±\sqrt{100+20\times 5}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-\left(-10\right)±\sqrt{100+100}}{2\left(-5\right)}
Multiply 20 times 5.
x=\frac{-\left(-10\right)±\sqrt{200}}{2\left(-5\right)}
Add 100 to 100.
x=\frac{-\left(-10\right)±10\sqrt{2}}{2\left(-5\right)}
Take the square root of 200.
x=\frac{10±10\sqrt{2}}{2\left(-5\right)}
The opposite of -10 is 10.
x=\frac{10±10\sqrt{2}}{-10}
Multiply 2 times -5.
x=\frac{10\sqrt{2}+10}{-10}
Now solve the equation x=\frac{10±10\sqrt{2}}{-10} when ± is plus. Add 10 to 10\sqrt{2}.
x=-\left(\sqrt{2}+1\right)
Divide 10+10\sqrt{2} by -10.
x=\frac{10-10\sqrt{2}}{-10}
Now solve the equation x=\frac{10±10\sqrt{2}}{-10} when ± is minus. Subtract 10\sqrt{2} from 10.
x=\sqrt{2}-1
Divide 10-10\sqrt{2} by -10.
x=-\left(\sqrt{2}+1\right) x=\sqrt{2}-1
The equation is now solved.
-5x^{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.
-5x^{2}-10x+5-5=-5
Subtract 5 from both sides of the equation.
-5x^{2}-10x=-5
Subtracting 5 from itself leaves 0.
\frac{-5x^{2}-10x}{-5}=-\frac{5}{-5}
Divide both sides by -5.
x^{2}+\left(-\frac{10}{-5}\right)x=-\frac{5}{-5}
Dividing by -5 undoes the multiplication by -5.
x^{2}+2x=-\frac{5}{-5}
Divide -10 by -5.
x^{2}+2x=1
Divide -5 by -5.
x^{2}+2x+1^{2}=1+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=1+1
Square 1.
x^{2}+2x+1=2
Add 1 to 1.
\left(x+1\right)^{2}=2
Factor x^{2}+2x+1. 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+1\right)^{2}}=\sqrt{2}
Take the square root of both sides of the equation.
x+1=\sqrt{2} x+1=-\sqrt{2}
Simplify.
x=\sqrt{2}-1 x=-\sqrt{2}-1
Subtract 1 from both sides of the equation.