Factor
\left(x-1\right)\left(2x+3\right)
Evaluate
\left(x-1\right)\left(2x+3\right)
Graph
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2x^{2}+x-3
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=1 ab=2\left(-3\right)=-6
Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-3. To find a and b, set up a system to be solved.
-1,6 -2,3
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 -6.
-1+6=5 -2+3=1
Calculate the sum for each pair.
a=-2 b=3
The solution is the pair that gives sum 1.
\left(2x^{2}-2x\right)+\left(3x-3\right)
Rewrite 2x^{2}+x-3 as \left(2x^{2}-2x\right)+\left(3x-3\right).
2x\left(x-1\right)+3\left(x-1\right)
Factor out 2x in the first and 3 in the second group.
\left(x-1\right)\left(2x+3\right)
Factor out common term x-1 by using distributive property.
2x^{2}+x-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.
x=\frac{-1±\sqrt{1^{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.
x=\frac{-1±\sqrt{1-4\times 2\left(-3\right)}}{2\times 2}
Square 1.
x=\frac{-1±\sqrt{1-8\left(-3\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-1±\sqrt{1+24}}{2\times 2}
Multiply -8 times -3.
x=\frac{-1±\sqrt{25}}{2\times 2}
Add 1 to 24.
x=\frac{-1±5}{2\times 2}
Take the square root of 25.
x=\frac{-1±5}{4}
Multiply 2 times 2.
x=\frac{4}{4}
Now solve the equation x=\frac{-1±5}{4} when ± is plus. Add -1 to 5.
x=1
Divide 4 by 4.
x=-\frac{6}{4}
Now solve the equation x=\frac{-1±5}{4} when ± is minus. Subtract 5 from -1.
x=-\frac{3}{2}
Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.
2x^{2}+x-3=2\left(x-1\right)\left(x-\left(-\frac{3}{2}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -\frac{3}{2} for x_{2}.
2x^{2}+x-3=2\left(x-1\right)\left(x+\frac{3}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
2x^{2}+x-3=2\left(x-1\right)\times \frac{2x+3}{2}
Add \frac{3}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
2x^{2}+x-3=\left(x-1\right)\left(2x+3\right)
Cancel out 2, the greatest common factor in 2 and 2.
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}