Solve for x
x=-\frac{1}{4}=-0.25
x=\frac{3}{4}=0.75
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x^{2}-\frac{1}{2}x-\frac{3}{16}=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\left(-\frac{3}{16}\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{1}{2} for b, and -\frac{3}{16} 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\left(-\frac{3}{16}\right)}}{2}
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+3}{4}}}{2}
Multiply -4 times -\frac{3}{16}.
x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{1}}{2}
Add \frac{1}{4} to \frac{3}{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)±1}{2}
Take the square root of 1.
x=\frac{\frac{1}{2}±1}{2}
The opposite of -\frac{1}{2} is \frac{1}{2}.
x=\frac{\frac{3}{2}}{2}
Now solve the equation x=\frac{\frac{1}{2}±1}{2} when ± is plus. Add \frac{1}{2} to 1.
x=\frac{3}{4}
Divide \frac{3}{2} by 2.
x=-\frac{\frac{1}{2}}{2}
Now solve the equation x=\frac{\frac{1}{2}±1}{2} when ± is minus. Subtract 1 from \frac{1}{2}.
x=-\frac{1}{4}
Divide -\frac{1}{2} by 2.
x=\frac{3}{4} x=-\frac{1}{4}
The equation is now solved.
x^{2}-\frac{1}{2}x-\frac{3}{16}=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.
x^{2}-\frac{1}{2}x-\frac{3}{16}-\left(-\frac{3}{16}\right)=-\left(-\frac{3}{16}\right)
Add \frac{3}{16} to both sides of the equation.
x^{2}-\frac{1}{2}x=-\left(-\frac{3}{16}\right)
Subtracting -\frac{3}{16} from itself leaves 0.
x^{2}-\frac{1}{2}x=\frac{3}{16}
Subtract -\frac{3}{16} from 0.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{3}{16}+\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{3+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{1}{4}
Add \frac{3}{16} 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{1}{4}
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{1}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{1}{2} x-\frac{1}{4}=-\frac{1}{2}
Simplify.
x=\frac{3}{4} x=-\frac{1}{4}
Add \frac{1}{4} 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
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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}