Solve for x
x=-2
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x^{2}+4x+4=0
Divide both sides by 2.
a+b=4 ab=1\times 4=4
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+4. To find a and b, set up a system to be solved.
1,4 2,2
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 4.
1+4=5 2+2=4
Calculate the sum for each pair.
a=2 b=2
The solution is the pair that gives sum 4.
\left(x^{2}+2x\right)+\left(2x+4\right)
Rewrite x^{2}+4x+4 as \left(x^{2}+2x\right)+\left(2x+4\right).
x\left(x+2\right)+2\left(x+2\right)
Factor out x in the first and 2 in the second group.
\left(x+2\right)\left(x+2\right)
Factor out common term x+2 by using distributive property.
\left(x+2\right)^{2}
Rewrite as a binomial square.
x=-2
To find equation solution, solve x+2=0.
2x^{2}+8x+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{-8±\sqrt{8^{2}-4\times 2\times 8}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-8±\sqrt{64-4\times 2\times 8}}{2\times 2}
Square 8.
x=\frac{-8±\sqrt{64-8\times 8}}{2\times 2}
Multiply -4 times 2.
x=\frac{-8±\sqrt{64-64}}{2\times 2}
Multiply -8 times 8.
x=\frac{-8±\sqrt{0}}{2\times 2}
Add 64 to -64.
x=-\frac{8}{2\times 2}
Take the square root of 0.
x=-\frac{8}{4}
Multiply 2 times 2.
x=-2
Divide -8 by 4.
2x^{2}+8x+8=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.
2x^{2}+8x+8-8=-8
Subtract 8 from both sides of the equation.
2x^{2}+8x=-8
Subtracting 8 from itself leaves 0.
\frac{2x^{2}+8x}{2}=-\frac{8}{2}
Divide both sides by 2.
x^{2}+\frac{8}{2}x=-\frac{8}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+4x=-\frac{8}{2}
Divide 8 by 2.
x^{2}+4x=-4
Divide -8 by 2.
x^{2}+4x+2^{2}=-4+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=-4+4
Square 2.
x^{2}+4x+4=0
Add -4 to 4.
\left(x+2\right)^{2}=0
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{0}
Take the square root of both sides of the equation.
x+2=0 x+2=0
Simplify.
x=-2 x=-2
Subtract 2 from both sides of the equation.
x=-2
The equation is now solved. Solutions are the same.
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}