Factor
\left(1-x\right)\left(3x+1\right)
Evaluate
\left(1-x\right)\left(3x+1\right)
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a+b=2 ab=-3=-3
Factor the expression by grouping. First, the expression needs to be rewritten as -3x^{2}+ax+bx+1. To find a and b, set up a system to be solved.
a=3 b=-1
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. The only such pair is the system solution.
\left(-3x^{2}+3x\right)+\left(-x+1\right)
Rewrite -3x^{2}+2x+1 as \left(-3x^{2}+3x\right)+\left(-x+1\right).
3x\left(-x+1\right)-x+1
Factor out 3x in -3x^{2}+3x.
\left(-x+1\right)\left(3x+1\right)
Factor out common term -x+1 by using distributive property.
-3x^{2}+2x+1=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{-2±\sqrt{2^{2}-4\left(-3\right)}}{2\left(-3\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{-2±\sqrt{4-4\left(-3\right)}}{2\left(-3\right)}
Square 2.
x=\frac{-2±\sqrt{4+12}}{2\left(-3\right)}
Multiply -4 times -3.
x=\frac{-2±\sqrt{16}}{2\left(-3\right)}
Add 4 to 12.
x=\frac{-2±4}{2\left(-3\right)}
Take the square root of 16.
x=\frac{-2±4}{-6}
Multiply 2 times -3.
x=\frac{2}{-6}
Now solve the equation x=\frac{-2±4}{-6} when ± is plus. Add -2 to 4.
x=-\frac{1}{3}
Reduce the fraction \frac{2}{-6} to lowest terms by extracting and canceling out 2.
x=-\frac{6}{-6}
Now solve the equation x=\frac{-2±4}{-6} when ± is minus. Subtract 4 from -2.
x=1
Divide -6 by -6.
-3x^{2}+2x+1=-3\left(x-\left(-\frac{1}{3}\right)\right)\left(x-1\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{3} for x_{1} and 1 for x_{2}.
-3x^{2}+2x+1=-3\left(x+\frac{1}{3}\right)\left(x-1\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
-3x^{2}+2x+1=-3\times \frac{-3x-1}{-3}\left(x-1\right)
Add \frac{1}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
-3x^{2}+2x+1=\left(-3x-1\right)\left(x-1\right)
Cancel out 3, the greatest common factor in -3 and 3.
x ^ 2 -\frac{2}{3}x -\frac{1}{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.
r + s = \frac{2}{3} rs = -\frac{1}{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{1}{3} - u s = \frac{1}{3} + u
Two numbers r and s sum up to \frac{2}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{2}{3} = \frac{1}{3}. 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}{3} - u) (\frac{1}{3} + u) = -\frac{1}{3}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{3}
\frac{1}{9} - u^2 = -\frac{1}{3}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{1}{3}-\frac{1}{9} = -\frac{4}{9}
Simplify the expression by subtracting \frac{1}{9} on both sides
u^2 = \frac{4}{9} u = \pm\sqrt{\frac{4}{9}} = \pm \frac{2}{3}
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}{3} - \frac{2}{3} = -0.333 s = \frac{1}{3} + \frac{2}{3} = 1
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}