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x^{2}+4x-84=62
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}+4x-84-62=62-62
Subtract 62 from both sides of the equation.
x^{2}+4x-84-62=0
Subtracting 62 from itself leaves 0.
x^{2}+4x-146=0
Subtract 62 from -84.
x=\frac{-4±\sqrt{4^{2}-4\left(-146\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and -146 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\left(-146\right)}}{2}
Square 4.
x=\frac{-4±\sqrt{16+584}}{2}
Multiply -4 times -146.
x=\frac{-4±\sqrt{600}}{2}
Add 16 to 584.
x=\frac{-4±10\sqrt{6}}{2}
Take the square root of 600.
x=\frac{10\sqrt{6}-4}{2}
Now solve the equation x=\frac{-4±10\sqrt{6}}{2} when ± is plus. Add -4 to 10\sqrt{6}.
x=5\sqrt{6}-2
Divide -4+10\sqrt{6} by 2.
x=\frac{-10\sqrt{6}-4}{2}
Now solve the equation x=\frac{-4±10\sqrt{6}}{2} when ± is minus. Subtract 10\sqrt{6} from -4.
x=-5\sqrt{6}-2
Divide -4-10\sqrt{6} by 2.
x=5\sqrt{6}-2 x=-5\sqrt{6}-2
The equation is now solved.
x^{2}+4x-84=62
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}+4x-84-\left(-84\right)=62-\left(-84\right)
Add 84 to both sides of the equation.
x^{2}+4x=62-\left(-84\right)
Subtracting -84 from itself leaves 0.
x^{2}+4x=146
Subtract -84 from 62.
x^{2}+4x+2^{2}=146+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=146+4
Square 2.
x^{2}+4x+4=150
Add 146 to 4.
\left(x+2\right)^{2}=150
Factor x^{2}+4x+4. 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+2\right)^{2}}=\sqrt{150}
Take the square root of both sides of the equation.
x+2=5\sqrt{6} x+2=-5\sqrt{6}
Simplify.
x=5\sqrt{6}-2 x=-5\sqrt{6}-2
Subtract 2 from both sides of the equation.