Solve for x (complex solution)
x=\frac{5+\sqrt{227}i}{18}\approx 0.277777778+0.837028843i
x=\frac{-\sqrt{227}i+5}{18}\approx 0.277777778-0.837028843i
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9x^{2}-5x+7=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(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 9\times 7}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -5 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-5\right)±\sqrt{25-4\times 9\times 7}}{2\times 9}
Square -5.
x=\frac{-\left(-5\right)±\sqrt{25-36\times 7}}{2\times 9}
Multiply -4 times 9.
x=\frac{-\left(-5\right)±\sqrt{25-252}}{2\times 9}
Multiply -36 times 7.
x=\frac{-\left(-5\right)±\sqrt{-227}}{2\times 9}
Add 25 to -252.
x=\frac{-\left(-5\right)±\sqrt{227}i}{2\times 9}
Take the square root of -227.
x=\frac{5±\sqrt{227}i}{2\times 9}
The opposite of -5 is 5.
x=\frac{5±\sqrt{227}i}{18}
Multiply 2 times 9.
x=\frac{5+\sqrt{227}i}{18}
Now solve the equation x=\frac{5±\sqrt{227}i}{18} when ± is plus. Add 5 to i\sqrt{227}.
x=\frac{-\sqrt{227}i+5}{18}
Now solve the equation x=\frac{5±\sqrt{227}i}{18} when ± is minus. Subtract i\sqrt{227} from 5.
x=\frac{5+\sqrt{227}i}{18} x=\frac{-\sqrt{227}i+5}{18}
The equation is now solved.
9x^{2}-5x+7=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.
9x^{2}-5x+7-7=-7
Subtract 7 from both sides of the equation.
9x^{2}-5x=-7
Subtracting 7 from itself leaves 0.
\frac{9x^{2}-5x}{9}=-\frac{7}{9}
Divide both sides by 9.
x^{2}-\frac{5}{9}x=-\frac{7}{9}
Dividing by 9 undoes the multiplication by 9.
x^{2}-\frac{5}{9}x+\left(-\frac{5}{18}\right)^{2}=-\frac{7}{9}+\left(-\frac{5}{18}\right)^{2}
Divide -\frac{5}{9}, the coefficient of the x term, by 2 to get -\frac{5}{18}. Then add the square of -\frac{5}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{5}{9}x+\frac{25}{324}=-\frac{7}{9}+\frac{25}{324}
Square -\frac{5}{18} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{5}{9}x+\frac{25}{324}=-\frac{227}{324}
Add -\frac{7}{9} to \frac{25}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{5}{18}\right)^{2}=-\frac{227}{324}
Factor x^{2}-\frac{5}{9}x+\frac{25}{324}. 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{5}{18}\right)^{2}}=\sqrt{-\frac{227}{324}}
Take the square root of both sides of the equation.
x-\frac{5}{18}=\frac{\sqrt{227}i}{18} x-\frac{5}{18}=-\frac{\sqrt{227}i}{18}
Simplify.
x=\frac{5+\sqrt{227}i}{18} x=\frac{-\sqrt{227}i+5}{18}
Add \frac{5}{18} to both sides of the equation.
x ^ 2 -\frac{5}{9}x +\frac{7}{9} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 9
r + s = \frac{5}{9} rs = \frac{7}{9}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = \frac{5}{18} - u s = \frac{5}{18} + u
Two numbers r and s sum up to \frac{5}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{5}{9} = \frac{5}{18}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(\frac{5}{18} - u) (\frac{5}{18} + u) = \frac{7}{9}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{7}{9}
\frac{25}{324} - u^2 = \frac{7}{9}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{7}{9}-\frac{25}{324} = \frac{227}{324}
Simplify the expression by subtracting \frac{25}{324} on both sides
u^2 = -\frac{227}{324} u = \pm\sqrt{-\frac{227}{324}} = \pm \frac{\sqrt{227}}{18}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{5}{18} - \frac{\sqrt{227}}{18}i = 0.278 - 0.837i s = \frac{5}{18} + \frac{\sqrt{227}}{18}i = 0.278 + 0.837i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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Linear equation
y = 3x + 4
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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
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