Solve for x
x=\frac{\sqrt{5}+1}{4}\approx 0.809016994
x=\frac{1-\sqrt{5}}{4}\approx -0.309016994
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4x^{2}-2x=1
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.
4x^{2}-2x-1=1-1
Subtract 1 from both sides of the equation.
4x^{2}-2x-1=0
Subtracting 1 from itself leaves 0.
x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 4\left(-1\right)}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -2 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2\right)±\sqrt{4-4\times 4\left(-1\right)}}{2\times 4}
Square -2.
x=\frac{-\left(-2\right)±\sqrt{4-16\left(-1\right)}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-2\right)±\sqrt{4+16}}{2\times 4}
Multiply -16 times -1.
x=\frac{-\left(-2\right)±\sqrt{20}}{2\times 4}
Add 4 to 16.
x=\frac{-\left(-2\right)±2\sqrt{5}}{2\times 4}
Take the square root of 20.
x=\frac{2±2\sqrt{5}}{2\times 4}
The opposite of -2 is 2.
x=\frac{2±2\sqrt{5}}{8}
Multiply 2 times 4.
x=\frac{2\sqrt{5}+2}{8}
Now solve the equation x=\frac{2±2\sqrt{5}}{8} when ± is plus. Add 2 to 2\sqrt{5}.
x=\frac{\sqrt{5}+1}{4}
Divide 2+2\sqrt{5} by 8.
x=\frac{2-2\sqrt{5}}{8}
Now solve the equation x=\frac{2±2\sqrt{5}}{8} when ± is minus. Subtract 2\sqrt{5} from 2.
x=\frac{1-\sqrt{5}}{4}
Divide 2-2\sqrt{5} by 8.
x=\frac{\sqrt{5}+1}{4} x=\frac{1-\sqrt{5}}{4}
The equation is now solved.
4x^{2}-2x=1
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{4x^{2}-2x}{4}=\frac{1}{4}
Divide both sides by 4.
x^{2}+\left(-\frac{2}{4}\right)x=\frac{1}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-\frac{1}{2}x=\frac{1}{4}
Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{1}{4}+\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}{4}+\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{5}{16}
Add \frac{1}{4} 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{5}{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{5}{16}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{\sqrt{5}}{4} x-\frac{1}{4}=-\frac{\sqrt{5}}{4}
Simplify.
x=\frac{\sqrt{5}+1}{4} x=\frac{1-\sqrt{5}}{4}
Add \frac{1}{4} to 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}