Solve for a
a=2\sqrt{2}+2\approx 4.828427125
a=2-2\sqrt{2}\approx -0.828427125
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-a^{2}+4a+4=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.
a=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 4}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-4±\sqrt{16-4\left(-1\right)\times 4}}{2\left(-1\right)}
Square 4.
a=\frac{-4±\sqrt{16+4\times 4}}{2\left(-1\right)}
Multiply -4 times -1.
a=\frac{-4±\sqrt{16+16}}{2\left(-1\right)}
Multiply 4 times 4.
a=\frac{-4±\sqrt{32}}{2\left(-1\right)}
Add 16 to 16.
a=\frac{-4±4\sqrt{2}}{2\left(-1\right)}
Take the square root of 32.
a=\frac{-4±4\sqrt{2}}{-2}
Multiply 2 times -1.
a=\frac{4\sqrt{2}-4}{-2}
Now solve the equation a=\frac{-4±4\sqrt{2}}{-2} when ± is plus. Add -4 to 4\sqrt{2}.
a=2-2\sqrt{2}
Divide -4+4\sqrt{2} by -2.
a=\frac{-4\sqrt{2}-4}{-2}
Now solve the equation a=\frac{-4±4\sqrt{2}}{-2} when ± is minus. Subtract 4\sqrt{2} from -4.
a=2\sqrt{2}+2
Divide -4-4\sqrt{2} by -2.
a=2-2\sqrt{2} a=2\sqrt{2}+2
The equation is now solved.
-a^{2}+4a+4=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.
-a^{2}+4a+4-4=-4
Subtract 4 from both sides of the equation.
-a^{2}+4a=-4
Subtracting 4 from itself leaves 0.
\frac{-a^{2}+4a}{-1}=-\frac{4}{-1}
Divide both sides by -1.
a^{2}+\frac{4}{-1}a=-\frac{4}{-1}
Dividing by -1 undoes the multiplication by -1.
a^{2}-4a=-\frac{4}{-1}
Divide 4 by -1.
a^{2}-4a=4
Divide -4 by -1.
a^{2}-4a+\left(-2\right)^{2}=4+\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.
a^{2}-4a+4=4+4
Square -2.
a^{2}-4a+4=8
Add 4 to 4.
\left(a-2\right)^{2}=8
Factor a^{2}-4a+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(a-2\right)^{2}}=\sqrt{8}
Take the square root of both sides of the equation.
a-2=2\sqrt{2} a-2=-2\sqrt{2}
Simplify.
a=2\sqrt{2}+2 a=2-2\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}