Solve for x
x=\frac{\sqrt{2}+1}{3}\approx 0.804737854
x=\frac{1-\sqrt{2}}{3}\approx -0.138071187
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\frac{3}{4}x^{2}-\frac{1}{2}x-\frac{1}{12}=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{-\left(-\frac{1}{2}\right)±\sqrt{\left(-\frac{1}{2}\right)^{2}-4\times \frac{3}{4}\left(-\frac{1}{12}\right)}}{2\times \frac{3}{4}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{4} for a, -\frac{1}{2} for b, and -\frac{1}{12} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-4\times \frac{3}{4}\left(-\frac{1}{12}\right)}}{2\times \frac{3}{4}}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-3\left(-\frac{1}{12}\right)}}{2\times \frac{3}{4}}
Multiply -4 times \frac{3}{4}.
x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1+1}{4}}}{2\times \frac{3}{4}}
Multiply -3 times -\frac{1}{12}.
x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{2}}}{2\times \frac{3}{4}}
Add \frac{1}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{-\left(-\frac{1}{2}\right)±\frac{\sqrt{2}}{2}}{2\times \frac{3}{4}}
Take the square root of \frac{1}{2}.
x=\frac{\frac{1}{2}±\frac{\sqrt{2}}{2}}{2\times \frac{3}{4}}
The opposite of -\frac{1}{2} is \frac{1}{2}.
x=\frac{\frac{1}{2}±\frac{\sqrt{2}}{2}}{\frac{3}{2}}
Multiply 2 times \frac{3}{4}.
x=\frac{\sqrt{2}+1}{\frac{3}{2}\times 2}
Now solve the equation x=\frac{\frac{1}{2}±\frac{\sqrt{2}}{2}}{\frac{3}{2}} when ± is plus. Add \frac{1}{2} to \frac{\sqrt{2}}{2}.
x=\frac{\sqrt{2}+1}{3}
Divide \frac{1+\sqrt{2}}{2} by \frac{3}{2} by multiplying \frac{1+\sqrt{2}}{2} by the reciprocal of \frac{3}{2}.
x=\frac{1-\sqrt{2}}{\frac{3}{2}\times 2}
Now solve the equation x=\frac{\frac{1}{2}±\frac{\sqrt{2}}{2}}{\frac{3}{2}} when ± is minus. Subtract \frac{\sqrt{2}}{2} from \frac{1}{2}.
x=\frac{1-\sqrt{2}}{3}
Divide \frac{1-\sqrt{2}}{2} by \frac{3}{2} by multiplying \frac{1-\sqrt{2}}{2} by the reciprocal of \frac{3}{2}.
x=\frac{\sqrt{2}+1}{3} x=\frac{1-\sqrt{2}}{3}
The equation is now solved.
\frac{3}{4}x^{2}-\frac{1}{2}x-\frac{1}{12}=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.
\frac{3}{4}x^{2}-\frac{1}{2}x-\frac{1}{12}-\left(-\frac{1}{12}\right)=-\left(-\frac{1}{12}\right)
Add \frac{1}{12} to both sides of the equation.
\frac{3}{4}x^{2}-\frac{1}{2}x=-\left(-\frac{1}{12}\right)
Subtracting -\frac{1}{12} from itself leaves 0.
\frac{3}{4}x^{2}-\frac{1}{2}x=\frac{1}{12}
Subtract -\frac{1}{12} from 0.
\frac{\frac{3}{4}x^{2}-\frac{1}{2}x}{\frac{3}{4}}=\frac{\frac{1}{12}}{\frac{3}{4}}
Divide both sides of the equation by \frac{3}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.
x^{2}+\left(-\frac{\frac{1}{2}}{\frac{3}{4}}\right)x=\frac{\frac{1}{12}}{\frac{3}{4}}
Dividing by \frac{3}{4} undoes the multiplication by \frac{3}{4}.
x^{2}-\frac{2}{3}x=\frac{\frac{1}{12}}{\frac{3}{4}}
Divide -\frac{1}{2} by \frac{3}{4} by multiplying -\frac{1}{2} by the reciprocal of \frac{3}{4}.
x^{2}-\frac{2}{3}x=\frac{1}{9}
Divide \frac{1}{12} by \frac{3}{4} by multiplying \frac{1}{12} by the reciprocal of \frac{3}{4}.
x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=\frac{1}{9}+\left(-\frac{1}{3}\right)^{2}
Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{1+1}{9}
Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{2}{9}
Add \frac{1}{9} to \frac{1}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{3}\right)^{2}=\frac{2}{9}
Factor x^{2}-\frac{2}{3}x+\frac{1}{9}. 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}{3}\right)^{2}}=\sqrt{\frac{2}{9}}
Take the square root of both sides of the equation.
x-\frac{1}{3}=\frac{\sqrt{2}}{3} x-\frac{1}{3}=-\frac{\sqrt{2}}{3}
Simplify.
x=\frac{\sqrt{2}+1}{3} x=\frac{1-\sqrt{2}}{3}
Add \frac{1}{3} to both sides of the equation.
Examples
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{ 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}