Solve for a
a=3
a=11
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a^{2}-14a+33=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-14 ab=33
To solve the equation, factor a^{2}-14a+33 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). To find a and b, set up a system to be solved.
-1,-33 -3,-11
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 33.
-1-33=-34 -3-11=-14
Calculate the sum for each pair.
a=-11 b=-3
The solution is the pair that gives sum -14.
\left(a-11\right)\left(a-3\right)
Rewrite factored expression \left(a+a\right)\left(a+b\right) using the obtained values.
a=11 a=3
To find equation solutions, solve a-11=0 and a-3=0.
a^{2}-14a+33=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-14 ab=1\times 33=33
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba+33. To find a and b, set up a system to be solved.
-1,-33 -3,-11
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 33.
-1-33=-34 -3-11=-14
Calculate the sum for each pair.
a=-11 b=-3
The solution is the pair that gives sum -14.
\left(a^{2}-11a\right)+\left(-3a+33\right)
Rewrite a^{2}-14a+33 as \left(a^{2}-11a\right)+\left(-3a+33\right).
a\left(a-11\right)-3\left(a-11\right)
Factor out a in the first and -3 in the second group.
\left(a-11\right)\left(a-3\right)
Factor out common term a-11 by using distributive property.
a=11 a=3
To find equation solutions, solve a-11=0 and a-3=0.
a^{2}-14a+33=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{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 33}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-\left(-14\right)±\sqrt{196-4\times 33}}{2}
Square -14.
a=\frac{-\left(-14\right)±\sqrt{196-132}}{2}
Multiply -4 times 33.
a=\frac{-\left(-14\right)±\sqrt{64}}{2}
Add 196 to -132.
a=\frac{-\left(-14\right)±8}{2}
Take the square root of 64.
a=\frac{14±8}{2}
The opposite of -14 is 14.
a=\frac{22}{2}
Now solve the equation a=\frac{14±8}{2} when ± is plus. Add 14 to 8.
a=11
Divide 22 by 2.
a=\frac{6}{2}
Now solve the equation a=\frac{14±8}{2} when ± is minus. Subtract 8 from 14.
a=3
Divide 6 by 2.
a=11 a=3
The equation is now solved.
a^{2}-14a+33=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}-14a+33-33=-33
Subtract 33 from both sides of the equation.
a^{2}-14a=-33
Subtracting 33 from itself leaves 0.
a^{2}-14a+\left(-7\right)^{2}=-33+\left(-7\right)^{2}
Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}-14a+49=-33+49
Square -7.
a^{2}-14a+49=16
Add -33 to 49.
\left(a-7\right)^{2}=16
Factor a^{2}-14a+49. 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-7\right)^{2}}=\sqrt{16}
Take the square root of both sides of the equation.
a-7=4 a-7=-4
Simplify.
a=11 a=3
Add 7 to both sides of the equation.
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Limits
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