Solve for x
x=\sqrt{2}+2\approx 3.414213562
x=2-\sqrt{2}\approx 0.585786438
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2x\left(4-x\right)=2\times 2
Multiply both sides by 2.
x\left(4-x\right)=2
Cancel out 2 on both sides.
4x-x^{2}=2
Use the distributive property to multiply x by 4-x.
4x-x^{2}-2=0
Subtract 2 from both sides.
-x^{2}+4x-2=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{-4±\sqrt{4^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}
Square 4.
x=\frac{-4±\sqrt{16+4\left(-2\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-4±\sqrt{16-8}}{2\left(-1\right)}
Multiply 4 times -2.
x=\frac{-4±\sqrt{8}}{2\left(-1\right)}
Add 16 to -8.
x=\frac{-4±2\sqrt{2}}{2\left(-1\right)}
Take the square root of 8.
x=\frac{-4±2\sqrt{2}}{-2}
Multiply 2 times -1.
x=\frac{2\sqrt{2}-4}{-2}
Now solve the equation x=\frac{-4±2\sqrt{2}}{-2} when ± is plus. Add -4 to 2\sqrt{2}.
x=2-\sqrt{2}
Divide -4+2\sqrt{2} by -2.
x=\frac{-2\sqrt{2}-4}{-2}
Now solve the equation x=\frac{-4±2\sqrt{2}}{-2} when ± is minus. Subtract 2\sqrt{2} from -4.
x=\sqrt{2}+2
Divide -4-2\sqrt{2} by -2.
x=2-\sqrt{2} x=\sqrt{2}+2
The equation is now solved.
2x\left(4-x\right)=2\times 2
Multiply both sides by 2.
x\left(4-x\right)=2
Cancel out 2 on both sides.
4x-x^{2}=2
Use the distributive property to multiply x by 4-x.
-x^{2}+4x=2
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}+4x}{-1}=\frac{2}{-1}
Divide both sides by -1.
x^{2}+\frac{4}{-1}x=\frac{2}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-4x=\frac{2}{-1}
Divide 4 by -1.
x^{2}-4x=-2
Divide 2 by -1.
x^{2}-4x+\left(-2\right)^{2}=-2+\left(-2\right)^{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=-2+4
Square -2.
x^{2}-4x+4=2
Add -2 to 4.
\left(x-2\right)^{2}=2
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{2}
Take the square root of both sides of the equation.
x-2=\sqrt{2} x-2=-\sqrt{2}
Simplify.
x=\sqrt{2}+2 x=2-\sqrt{2}
Add 2 to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}