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4x-\left(x^{2}-4\right)=2\left(x-2\right)
Consider \left(x-2\right)\left(x+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.
4x-x^{2}+4=2\left(x-2\right)
To find the opposite of x^{2}-4, find the opposite of each term.
4x-x^{2}+4=2x-4
Use the distributive property to multiply 2 by x-2.
4x-x^{2}+4-2x=-4
Subtract 2x from both sides.
2x-x^{2}+4=-4
Combine 4x and -2x to get 2x.
2x-x^{2}+4+4=0
Add 4 to both sides.
2x-x^{2}+8=0
Add 4 and 4 to get 8.
-x^{2}+2x+8=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{-2±\sqrt{2^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-1\right)\times 8}}{2\left(-1\right)}
Square 2.
x=\frac{-2±\sqrt{4+4\times 8}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-2±\sqrt{4+32}}{2\left(-1\right)}
Multiply 4 times 8.
x=\frac{-2±\sqrt{36}}{2\left(-1\right)}
Add 4 to 32.
x=\frac{-2±6}{2\left(-1\right)}
Take the square root of 36.
x=\frac{-2±6}{-2}
Multiply 2 times -1.
x=\frac{4}{-2}
Now solve the equation x=\frac{-2±6}{-2} when ± is plus. Add -2 to 6.
x=-2
Divide 4 by -2.
x=-\frac{8}{-2}
Now solve the equation x=\frac{-2±6}{-2} when ± is minus. Subtract 6 from -2.
x=4
Divide -8 by -2.
x=-2 x=4
The equation is now solved.
4x-\left(x^{2}-4\right)=2\left(x-2\right)
Consider \left(x-2\right)\left(x+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.
4x-x^{2}+4=2\left(x-2\right)
To find the opposite of x^{2}-4, find the opposite of each term.
4x-x^{2}+4=2x-4
Use the distributive property to multiply 2 by x-2.
4x-x^{2}+4-2x=-4
Subtract 2x from both sides.
2x-x^{2}+4=-4
Combine 4x and -2x to get 2x.
2x-x^{2}=-4-4
Subtract 4 from both sides.
2x-x^{2}=-8
Subtract 4 from -4 to get -8.
-x^{2}+2x=-8
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.
\frac{-x^{2}+2x}{-1}=-\frac{8}{-1}
Divide both sides by -1.
x^{2}+\frac{2}{-1}x=-\frac{8}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-2x=-\frac{8}{-1}
Divide 2 by -1.
x^{2}-2x=8
Divide -8 by -1.
x^{2}-2x+1=8+1
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=9
Add 8 to 1.
\left(x-1\right)^{2}=9
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{9}
Take the square root of both sides of the equation.
x-1=3 x-1=-3
Simplify.
x=4 x=-2
Add 1 to both sides of the equation.