Factor
\left(s-7\right)\left(2s+1\right)
Evaluate
\left(s-7\right)\left(2s+1\right)
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a+b=-13 ab=2\left(-7\right)=-14
Factor the expression by grouping. First, the expression needs to be rewritten as 2s^{2}+as+bs-7. To find a and b, set up a system to be solved.
1,-14 2,-7
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 -14.
1-14=-13 2-7=-5
Calculate the sum for each pair.
a=-14 b=1
The solution is the pair that gives sum -13.
\left(2s^{2}-14s\right)+\left(s-7\right)
Rewrite 2s^{2}-13s-7 as \left(2s^{2}-14s\right)+\left(s-7\right).
2s\left(s-7\right)+s-7
Factor out 2s in 2s^{2}-14s.
\left(s-7\right)\left(2s+1\right)
Factor out common term s-7 by using distributive property.
2s^{2}-13s-7=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.
s=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 2\left(-7\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.
s=\frac{-\left(-13\right)±\sqrt{169-4\times 2\left(-7\right)}}{2\times 2}
Square -13.
s=\frac{-\left(-13\right)±\sqrt{169-8\left(-7\right)}}{2\times 2}
Multiply -4 times 2.
s=\frac{-\left(-13\right)±\sqrt{169+56}}{2\times 2}
Multiply -8 times -7.
s=\frac{-\left(-13\right)±\sqrt{225}}{2\times 2}
Add 169 to 56.
s=\frac{-\left(-13\right)±15}{2\times 2}
Take the square root of 225.
s=\frac{13±15}{2\times 2}
The opposite of -13 is 13.
s=\frac{13±15}{4}
Multiply 2 times 2.
s=\frac{28}{4}
Now solve the equation s=\frac{13±15}{4} when ± is plus. Add 13 to 15.
s=7
Divide 28 by 4.
s=-\frac{2}{4}
Now solve the equation s=\frac{13±15}{4} when ± is minus. Subtract 15 from 13.
s=-\frac{1}{2}
Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.
2s^{2}-13s-7=2\left(s-7\right)\left(s-\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 7 for x_{1} and -\frac{1}{2} for x_{2}.
2s^{2}-13s-7=2\left(s-7\right)\left(s+\frac{1}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
2s^{2}-13s-7=2\left(s-7\right)\times \frac{2s+1}{2}
Add \frac{1}{2} to s by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
2s^{2}-13s-7=\left(s-7\right)\left(2s+1\right)
Cancel out 2, the greatest common factor in 2 and 2.
x ^ 2 -\frac{13}{2}x -\frac{7}{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{13}{2} rs = -\frac{7}{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{13}{4} - u s = \frac{13}{4} + u
Two numbers r and s sum up to \frac{13}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{13}{2} = \frac{13}{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{13}{4} - u) (\frac{13}{4} + u) = -\frac{7}{2}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{2}
\frac{169}{16} - u^2 = -\frac{7}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{7}{2}-\frac{169}{16} = -\frac{225}{16}
Simplify the expression by subtracting \frac{169}{16} on both sides
u^2 = \frac{225}{16} u = \pm\sqrt{\frac{225}{16}} = \pm \frac{15}{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{13}{4} - \frac{15}{4} = -0.500 s = \frac{13}{4} + \frac{15}{4} = 7
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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