Solve for λ
\lambda =-1
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\lambda ^{2}+2\lambda +1=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\lambda +1\right)^{2}.
a+b=2 ab=1
To solve the equation, factor \lambda ^{2}+2\lambda +1 using formula \lambda ^{2}+\left(a+b\right)\lambda +ab=\left(\lambda +a\right)\left(\lambda +b\right). To find a and b, set up a system to be solved.
a=1 b=1
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.
\left(\lambda +1\right)\left(\lambda +1\right)
Rewrite factored expression \left(\lambda +a\right)\left(\lambda +b\right) using the obtained values.
\left(\lambda +1\right)^{2}
Rewrite as a binomial square.
\lambda =-1
To find equation solution, solve \lambda +1=0.
\lambda ^{2}+2\lambda +1=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\lambda +1\right)^{2}.
a+b=2 ab=1\times 1=1
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as \lambda ^{2}+a\lambda +b\lambda +1. To find a and b, set up a system to be solved.
a=1 b=1
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.
\left(\lambda ^{2}+\lambda \right)+\left(\lambda +1\right)
Rewrite \lambda ^{2}+2\lambda +1 as \left(\lambda ^{2}+\lambda \right)+\left(\lambda +1\right).
\lambda \left(\lambda +1\right)+\lambda +1
Factor out \lambda in \lambda ^{2}+\lambda .
\left(\lambda +1\right)\left(\lambda +1\right)
Factor out common term \lambda +1 by using distributive property.
\left(\lambda +1\right)^{2}
Rewrite as a binomial square.
\lambda =-1
To find equation solution, solve \lambda +1=0.
\lambda ^{2}+2\lambda +1=0
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\lambda +1\right)^{2}.
\lambda =\frac{-2±\sqrt{2^{2}-4}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
\lambda =\frac{-2±\sqrt{4-4}}{2}
Square 2.
\lambda =\frac{-2±\sqrt{0}}{2}
Add 4 to -4.
\lambda =-\frac{2}{2}
Take the square root of 0.
\lambda =-1
Divide -2 by 2.
\sqrt{\left(\lambda +1\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
\lambda +1=0 \lambda +1=0
Simplify.
\lambda =-1 \lambda =-1
Subtract 1 from both sides of the equation.
\lambda =-1
The equation is now solved. Solutions are the same.
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Limits
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