Factor
\left(b-3\right)\left(2b+1\right)
Evaluate
\left(b-3\right)\left(2b+1\right)
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p+q=-5 pq=2\left(-3\right)=-6
Factor the expression by grouping. First, the expression needs to be rewritten as 2b^{2}+pb+qb-3. To find p and q, set up a system to be solved.
1,-6 2,-3
Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.
1-6=-5 2-3=-1
Calculate the sum for each pair.
p=-6 q=1
The solution is the pair that gives sum -5.
\left(2b^{2}-6b\right)+\left(b-3\right)
Rewrite 2b^{2}-5b-3 as \left(2b^{2}-6b\right)+\left(b-3\right).
2b\left(b-3\right)+b-3
Factor out 2b in 2b^{2}-6b.
\left(b-3\right)\left(2b+1\right)
Factor out common term b-3 by using distributive property.
2b^{2}-5b-3=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.
b=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\left(-3\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.
b=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-3\right)}}{2\times 2}
Square -5.
b=\frac{-\left(-5\right)±\sqrt{25-8\left(-3\right)}}{2\times 2}
Multiply -4 times 2.
b=\frac{-\left(-5\right)±\sqrt{25+24}}{2\times 2}
Multiply -8 times -3.
b=\frac{-\left(-5\right)±\sqrt{49}}{2\times 2}
Add 25 to 24.
b=\frac{-\left(-5\right)±7}{2\times 2}
Take the square root of 49.
b=\frac{5±7}{2\times 2}
The opposite of -5 is 5.
b=\frac{5±7}{4}
Multiply 2 times 2.
b=\frac{12}{4}
Now solve the equation b=\frac{5±7}{4} when ± is plus. Add 5 to 7.
b=3
Divide 12 by 4.
b=-\frac{2}{4}
Now solve the equation b=\frac{5±7}{4} when ± is minus. Subtract 7 from 5.
b=-\frac{1}{2}
Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.
2b^{2}-5b-3=2\left(b-3\right)\left(b-\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 3 for x_{1} and -\frac{1}{2} for x_{2}.
2b^{2}-5b-3=2\left(b-3\right)\left(b+\frac{1}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
2b^{2}-5b-3=2\left(b-3\right)\times \frac{2b+1}{2}
Add \frac{1}{2} to b by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
2b^{2}-5b-3=\left(b-3\right)\left(2b+1\right)
Cancel out 2, the greatest common factor in 2 and 2.
x ^ 2 -\frac{5}{2}x -\frac{3}{2} = 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{5}{2} rs = -\frac{3}{2}
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{5}{4} - u s = \frac{5}{4} + u
Two numbers r and s sum up to \frac{5}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{5}{2} = \frac{5}{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{5}{4} - u) (\frac{5}{4} + u) = -\frac{3}{2}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{2}
\frac{25}{16} - u^2 = -\frac{3}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{3}{2}-\frac{25}{16} = -\frac{49}{16}
Simplify the expression by subtracting \frac{25}{16} on both sides
u^2 = \frac{49}{16} u = \pm\sqrt{\frac{49}{16}} = \pm \frac{7}{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{5}{4} - \frac{7}{4} = -0.500 s = \frac{5}{4} + \frac{7}{4} = 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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Simultaneous equation
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Limits
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