Factor
-\left(x-4\right)\left(x+3\right)
Evaluate
-\left(x-4\right)\left(x+3\right)
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a+b=1 ab=-12=-12
Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+12. 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 positive, the positive number has greater absolute value than the negative. 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=4 b=-3
The solution is the pair that gives sum 1.
\left(-x^{2}+4x\right)+\left(-3x+12\right)
Rewrite -x^{2}+x+12 as \left(-x^{2}+4x\right)+\left(-3x+12\right).
-x\left(x-4\right)-3\left(x-4\right)
Factor out -x in the first and -3 in the second group.
\left(x-4\right)\left(-x-3\right)
Factor out common term x-4 by using distributive property.
-x^{2}+x+12=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.
x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}
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{-1±\sqrt{1-4\left(-1\right)\times 12}}{2\left(-1\right)}
Square 1.
x=\frac{-1±\sqrt{1+4\times 12}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-1±\sqrt{1+48}}{2\left(-1\right)}
Multiply 4 times 12.
x=\frac{-1±\sqrt{49}}{2\left(-1\right)}
Add 1 to 48.
x=\frac{-1±7}{2\left(-1\right)}
Take the square root of 49.
x=\frac{-1±7}{-2}
Multiply 2 times -1.
x=\frac{6}{-2}
Now solve the equation x=\frac{-1±7}{-2} when ± is plus. Add -1 to 7.
x=-3
Divide 6 by -2.
x=-\frac{8}{-2}
Now solve the equation x=\frac{-1±7}{-2} when ± is minus. Subtract 7 from -1.
x=4
Divide -8 by -2.
-x^{2}+x+12=-\left(x-\left(-3\right)\right)\left(x-4\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 4 for x_{2}.
-x^{2}+x+12=-\left(x+3\right)\left(x-4\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -1x -12 = 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.
r + s = 1 rs = -12
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{1}{2} - u s = \frac{1}{2} + u
Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{2}. 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{1}{2} - u) (\frac{1}{2} + u) = -12
To solve for unknown quantity u, substitute these in the product equation rs = -12
\frac{1}{4} - u^2 = -12
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -12-\frac{1}{4} = -\frac{49}{4}
Simplify the expression by subtracting \frac{1}{4} on both sides
u^2 = \frac{49}{4} u = \pm\sqrt{\frac{49}{4}} = \pm \frac{7}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{1}{2} - \frac{7}{2} = -3 s = \frac{1}{2} + \frac{7}{2} = 4
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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\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
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Limits
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