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Solve for x (complex solution)
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x^{2}+6x-4=30
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^{2}+6x-4-30=30-30
Subtract 30 from both sides of the equation.
x^{2}+6x-4-30=0
Subtracting 30 from itself leaves 0.
x^{2}+6x-34=0
Subtract 30 from -4.
x=\frac{-6±\sqrt{6^{2}-4\left(-34\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\left(-34\right)}}{2}
Square 6.
x=\frac{-6±\sqrt{36+136}}{2}
Multiply -4 times -34.
x=\frac{-6±\sqrt{172}}{2}
Add 36 to 136.
x=\frac{-6±2\sqrt{43}}{2}
Take the square root of 172.
x=\frac{2\sqrt{43}-6}{2}
Now solve the equation x=\frac{-6±2\sqrt{43}}{2} when ± is plus. Add -6 to 2\sqrt{43}.
x=\sqrt{43}-3
Divide -6+2\sqrt{43} by 2.
x=\frac{-2\sqrt{43}-6}{2}
Now solve the equation x=\frac{-6±2\sqrt{43}}{2} when ± is minus. Subtract 2\sqrt{43} from -6.
x=-\sqrt{43}-3
Divide -6-2\sqrt{43} by 2.
x=\sqrt{43}-3 x=-\sqrt{43}-3
The equation is now solved.
x^{2}+6x-4=30
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.
x^{2}+6x-4-\left(-4\right)=30-\left(-4\right)
Add 4 to both sides of the equation.
x^{2}+6x=30-\left(-4\right)
Subtracting -4 from itself leaves 0.
x^{2}+6x=34
Subtract -4 from 30.
x^{2}+6x+3^{2}=34+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=34+9
Square 3.
x^{2}+6x+9=43
Add 34 to 9.
\left(x+3\right)^{2}=43
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{43}
Take the square root of both sides of the equation.
x+3=\sqrt{43} x+3=-\sqrt{43}
Simplify.
x=\sqrt{43}-3 x=-\sqrt{43}-3
Subtract 3 from both sides of the equation.
x^{2}+6x-4=30
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^{2}+6x-4-30=30-30
Subtract 30 from both sides of the equation.
x^{2}+6x-4-30=0
Subtracting 30 from itself leaves 0.
x^{2}+6x-34=0
Subtract 30 from -4.
x=\frac{-6±\sqrt{6^{2}-4\left(-34\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\left(-34\right)}}{2}
Square 6.
x=\frac{-6±\sqrt{36+136}}{2}
Multiply -4 times -34.
x=\frac{-6±\sqrt{172}}{2}
Add 36 to 136.
x=\frac{-6±2\sqrt{43}}{2}
Take the square root of 172.
x=\frac{2\sqrt{43}-6}{2}
Now solve the equation x=\frac{-6±2\sqrt{43}}{2} when ± is plus. Add -6 to 2\sqrt{43}.
x=\sqrt{43}-3
Divide -6+2\sqrt{43} by 2.
x=\frac{-2\sqrt{43}-6}{2}
Now solve the equation x=\frac{-6±2\sqrt{43}}{2} when ± is minus. Subtract 2\sqrt{43} from -6.
x=-\sqrt{43}-3
Divide -6-2\sqrt{43} by 2.
x=\sqrt{43}-3 x=-\sqrt{43}-3
The equation is now solved.
x^{2}+6x-4=30
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.
x^{2}+6x-4-\left(-4\right)=30-\left(-4\right)
Add 4 to both sides of the equation.
x^{2}+6x=30-\left(-4\right)
Subtracting -4 from itself leaves 0.
x^{2}+6x=34
Subtract -4 from 30.
x^{2}+6x+3^{2}=34+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=34+9
Square 3.
x^{2}+6x+9=43
Add 34 to 9.
\left(x+3\right)^{2}=43
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{43}
Take the square root of both sides of the equation.
x+3=\sqrt{43} x+3=-\sqrt{43}
Simplify.
x=\sqrt{43}-3 x=-\sqrt{43}-3
Subtract 3 from both sides of the equation.