Solve for x
x=-\frac{2}{11}\approx -0.181818182
x=3
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a+b=-31 ab=11\left(-6\right)=-66
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 11x^{2}+ax+bx-6. To find a and b, set up a system to be solved.
1,-66 2,-33 3,-22 6,-11
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -66.
1-66=-65 2-33=-31 3-22=-19 6-11=-5
Calculate the sum for each pair.
a=-33 b=2
The solution is the pair that gives sum -31.
\left(11x^{2}-33x\right)+\left(2x-6\right)
Rewrite 11x^{2}-31x-6 as \left(11x^{2}-33x\right)+\left(2x-6\right).
11x\left(x-3\right)+2\left(x-3\right)
Factor out 11x in the first and 2 in the second group.
\left(x-3\right)\left(11x+2\right)
Factor out common term x-3 by using distributive property.
x=3 x=-\frac{2}{11}
To find equation solutions, solve x-3=0 and 11x+2=0.
11x^{2}-31x-6=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{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\times 11\left(-6\right)}}{2\times 11}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, -31 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-31\right)±\sqrt{961-4\times 11\left(-6\right)}}{2\times 11}
Square -31.
x=\frac{-\left(-31\right)±\sqrt{961-44\left(-6\right)}}{2\times 11}
Multiply -4 times 11.
x=\frac{-\left(-31\right)±\sqrt{961+264}}{2\times 11}
Multiply -44 times -6.
x=\frac{-\left(-31\right)±\sqrt{1225}}{2\times 11}
Add 961 to 264.
x=\frac{-\left(-31\right)±35}{2\times 11}
Take the square root of 1225.
x=\frac{31±35}{2\times 11}
The opposite of -31 is 31.
x=\frac{31±35}{22}
Multiply 2 times 11.
x=\frac{66}{22}
Now solve the equation x=\frac{31±35}{22} when ± is plus. Add 31 to 35.
x=3
Divide 66 by 22.
x=-\frac{4}{22}
Now solve the equation x=\frac{31±35}{22} when ± is minus. Subtract 35 from 31.
x=-\frac{2}{11}
Reduce the fraction \frac{-4}{22} to lowest terms by extracting and canceling out 2.
x=3 x=-\frac{2}{11}
The equation is now solved.
11x^{2}-31x-6=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.
11x^{2}-31x-6-\left(-6\right)=-\left(-6\right)
Add 6 to both sides of the equation.
11x^{2}-31x=-\left(-6\right)
Subtracting -6 from itself leaves 0.
11x^{2}-31x=6
Subtract -6 from 0.
\frac{11x^{2}-31x}{11}=\frac{6}{11}
Divide both sides by 11.
x^{2}-\frac{31}{11}x=\frac{6}{11}
Dividing by 11 undoes the multiplication by 11.
x^{2}-\frac{31}{11}x+\left(-\frac{31}{22}\right)^{2}=\frac{6}{11}+\left(-\frac{31}{22}\right)^{2}
Divide -\frac{31}{11}, the coefficient of the x term, by 2 to get -\frac{31}{22}. Then add the square of -\frac{31}{22} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{31}{11}x+\frac{961}{484}=\frac{6}{11}+\frac{961}{484}
Square -\frac{31}{22} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{31}{11}x+\frac{961}{484}=\frac{1225}{484}
Add \frac{6}{11} to \frac{961}{484} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{31}{22}\right)^{2}=\frac{1225}{484}
Factor x^{2}-\frac{31}{11}x+\frac{961}{484}. 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-\frac{31}{22}\right)^{2}}=\sqrt{\frac{1225}{484}}
Take the square root of both sides of the equation.
x-\frac{31}{22}=\frac{35}{22} x-\frac{31}{22}=-\frac{35}{22}
Simplify.
x=3 x=-\frac{2}{11}
Add \frac{31}{22} to both sides of the equation.
x ^ 2 -\frac{31}{11}x -\frac{6}{11} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 11
r + s = \frac{31}{11} rs = -\frac{6}{11}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = \frac{31}{22} - u s = \frac{31}{22} + u
Two numbers r and s sum up to \frac{31}{11} exactly when the average of the two numbers is \frac{1}{2}*\frac{31}{11} = \frac{31}{22}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(\frac{31}{22} - u) (\frac{31}{22} + u) = -\frac{6}{11}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{6}{11}
\frac{961}{484} - u^2 = -\frac{6}{11}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{6}{11}-\frac{961}{484} = -\frac{1225}{484}
Simplify the expression by subtracting \frac{961}{484} on both sides
u^2 = \frac{1225}{484} u = \pm\sqrt{\frac{1225}{484}} = \pm \frac{35}{22}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{31}{22} - \frac{35}{22} = -0.182 s = \frac{31}{22} + \frac{35}{22} = 3
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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