Solve for q
q=9
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q^{2}-18q+81=0
Add 81 to both sides.
a+b=-18 ab=81
To solve the equation, factor q^{2}-18q+81 using formula q^{2}+\left(a+b\right)q+ab=\left(q+a\right)\left(q+b\right). To find a and b, set up a system to be solved.
-1,-81 -3,-27 -9,-9
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 81.
-1-81=-82 -3-27=-30 -9-9=-18
Calculate the sum for each pair.
a=-9 b=-9
The solution is the pair that gives sum -18.
\left(q-9\right)\left(q-9\right)
Rewrite factored expression \left(q+a\right)\left(q+b\right) using the obtained values.
\left(q-9\right)^{2}
Rewrite as a binomial square.
q=9
To find equation solution, solve q-9=0.
q^{2}-18q+81=0
Add 81 to both sides.
a+b=-18 ab=1\times 81=81
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as q^{2}+aq+bq+81. To find a and b, set up a system to be solved.
-1,-81 -3,-27 -9,-9
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 81.
-1-81=-82 -3-27=-30 -9-9=-18
Calculate the sum for each pair.
a=-9 b=-9
The solution is the pair that gives sum -18.
\left(q^{2}-9q\right)+\left(-9q+81\right)
Rewrite q^{2}-18q+81 as \left(q^{2}-9q\right)+\left(-9q+81\right).
q\left(q-9\right)-9\left(q-9\right)
Factor out q in the first and -9 in the second group.
\left(q-9\right)\left(q-9\right)
Factor out common term q-9 by using distributive property.
\left(q-9\right)^{2}
Rewrite as a binomial square.
q=9
To find equation solution, solve q-9=0.
q^{2}-18q=-81
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.
q^{2}-18q-\left(-81\right)=-81-\left(-81\right)
Add 81 to both sides of the equation.
q^{2}-18q-\left(-81\right)=0
Subtracting -81 from itself leaves 0.
q^{2}-18q+81=0
Subtract -81 from 0.
q=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 81}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
q=\frac{-\left(-18\right)±\sqrt{324-4\times 81}}{2}
Square -18.
q=\frac{-\left(-18\right)±\sqrt{324-324}}{2}
Multiply -4 times 81.
q=\frac{-\left(-18\right)±\sqrt{0}}{2}
Add 324 to -324.
q=-\frac{-18}{2}
Take the square root of 0.
q=\frac{18}{2}
The opposite of -18 is 18.
q=9
Divide 18 by 2.
q^{2}-18q=-81
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.
q^{2}-18q+\left(-9\right)^{2}=-81+\left(-9\right)^{2}
Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
q^{2}-18q+81=-81+81
Square -9.
q^{2}-18q+81=0
Add -81 to 81.
\left(q-9\right)^{2}=0
Factor q^{2}-18q+81. 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(q-9\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
q-9=0 q-9=0
Simplify.
q=9 q=9
Add 9 to both sides of the equation.
q=9
The equation is now solved. Solutions are the same.
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Simultaneous equation
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Limits
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