Factor
\left(d-6\right)\left(2d+1\right)
Evaluate
\left(d-6\right)\left(2d+1\right)
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a+b=-11 ab=2\left(-6\right)=-12
Factor the expression by grouping. First, the expression needs to be rewritten as 2d^{2}+ad+bd-6. To find a and b, set up a system to be solved.
1,-12 2,-6 3,-4
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 -12.
1-12=-11 2-6=-4 3-4=-1
Calculate the sum for each pair.
a=-12 b=1
The solution is the pair that gives sum -11.
\left(2d^{2}-12d\right)+\left(d-6\right)
Rewrite 2d^{2}-11d-6 as \left(2d^{2}-12d\right)+\left(d-6\right).
2d\left(d-6\right)+d-6
Factor out 2d in 2d^{2}-12d.
\left(d-6\right)\left(2d+1\right)
Factor out common term d-6 by using distributive property.
2d^{2}-11d-6=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
d=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\left(-6\right)}}{2\times 2}
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.
d=\frac{-\left(-11\right)±\sqrt{121-4\times 2\left(-6\right)}}{2\times 2}
Square -11.
d=\frac{-\left(-11\right)±\sqrt{121-8\left(-6\right)}}{2\times 2}
Multiply -4 times 2.
d=\frac{-\left(-11\right)±\sqrt{121+48}}{2\times 2}
Multiply -8 times -6.
d=\frac{-\left(-11\right)±\sqrt{169}}{2\times 2}
Add 121 to 48.
d=\frac{-\left(-11\right)±13}{2\times 2}
Take the square root of 169.
d=\frac{11±13}{2\times 2}
The opposite of -11 is 11.
d=\frac{11±13}{4}
Multiply 2 times 2.
d=\frac{24}{4}
Now solve the equation d=\frac{11±13}{4} when ± is plus. Add 11 to 13.
d=6
Divide 24 by 4.
d=-\frac{2}{4}
Now solve the equation d=\frac{11±13}{4} when ± is minus. Subtract 13 from 11.
d=-\frac{1}{2}
Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.
2d^{2}-11d-6=2\left(d-6\right)\left(d-\left(-\frac{1}{2}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -\frac{1}{2} for x_{2}.
2d^{2}-11d-6=2\left(d-6\right)\left(d+\frac{1}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
2d^{2}-11d-6=2\left(d-6\right)\times \frac{2d+1}{2}
Add \frac{1}{2} to d by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
2d^{2}-11d-6=\left(d-6\right)\left(2d+1\right)
Cancel out 2, the greatest common factor in 2 and 2.
x ^ 2 -\frac{11}{2}x -3 = 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 2
r + s = \frac{11}{2} rs = -3
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{11}{4} - u s = \frac{11}{4} + u
Two numbers r and s sum up to \frac{11}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{11}{2} = \frac{11}{4}. 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{11}{4} - u) (\frac{11}{4} + u) = -3
To solve for unknown quantity u, substitute these in the product equation rs = -3
\frac{121}{16} - u^2 = -3
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -3-\frac{121}{16} = -\frac{169}{16}
Simplify the expression by subtracting \frac{121}{16} on both sides
u^2 = \frac{169}{16} u = \pm\sqrt{\frac{169}{16}} = \pm \frac{13}{4}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{11}{4} - \frac{13}{4} = -0.500 s = \frac{11}{4} + \frac{13}{4} = 6
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
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Simultaneous equation
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Integration
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Limits
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