Solve for x
x=-1
x=\frac{1}{2}=0.5
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a+b=1 ab=2\left(-1\right)=-2
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-1. To find a and b, set up a system to be solved.
a=-1 b=2
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(2x^{2}-x\right)+\left(2x-1\right)
Rewrite 2x^{2}+x-1 as \left(2x^{2}-x\right)+\left(2x-1\right).
x\left(2x-1\right)+2x-1
Factor out x in 2x^{2}-x.
\left(2x-1\right)\left(x+1\right)
Factor out common term 2x-1 by using distributive property.
x=\frac{1}{2} x=-1
To find equation solutions, solve 2x-1=0 and x+1=0.
2x^{2}+x-1=0
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^{2}-4\times 2\left(-1\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 1 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\times 2\left(-1\right)}}{2\times 2}
Square 1.
x=\frac{-1±\sqrt{1-8\left(-1\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-1±\sqrt{1+8}}{2\times 2}
Multiply -8 times -1.
x=\frac{-1±\sqrt{9}}{2\times 2}
Add 1 to 8.
x=\frac{-1±3}{2\times 2}
Take the square root of 9.
x=\frac{-1±3}{4}
Multiply 2 times 2.
x=\frac{2}{4}
Now solve the equation x=\frac{-1±3}{4} when ± is plus. Add -1 to 3.
x=\frac{1}{2}
Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.
x=-\frac{4}{4}
Now solve the equation x=\frac{-1±3}{4} when ± is minus. Subtract 3 from -1.
x=-1
Divide -4 by 4.
x=\frac{1}{2} x=-1
The equation is now solved.
2x^{2}+x-1=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
2x^{2}+x-1-\left(-1\right)=-\left(-1\right)
Add 1 to both sides of the equation.
2x^{2}+x=-\left(-1\right)
Subtracting -1 from itself leaves 0.
2x^{2}+x=1
Subtract -1 from 0.
\frac{2x^{2}+x}{2}=\frac{1}{2}
Divide both sides by 2.
x^{2}+\frac{1}{2}x=\frac{1}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=\frac{1}{2}+\left(\frac{1}{4}\right)^{2}
Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{1}{2}+\frac{1}{16}
Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{9}{16}
Add \frac{1}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{1}{4}\right)^{2}=\frac{9}{16}
Factor x^{2}+\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{9}{16}}
Take the square root of both sides of the equation.
x+\frac{1}{4}=\frac{3}{4} x+\frac{1}{4}=-\frac{3}{4}
Simplify.
x=\frac{1}{2} x=-1
Subtract \frac{1}{4} from both sides of the equation.
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}