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5x^{2}-9x+3=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(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 5\times 3}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -9 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-9\right)±\sqrt{81-4\times 5\times 3}}{2\times 5}
Square -9.
x=\frac{-\left(-9\right)±\sqrt{81-20\times 3}}{2\times 5}
Multiply -4 times 5.
x=\frac{-\left(-9\right)±\sqrt{81-60}}{2\times 5}
Multiply -20 times 3.
x=\frac{-\left(-9\right)±\sqrt{21}}{2\times 5}
Add 81 to -60.
x=\frac{9±\sqrt{21}}{2\times 5}
The opposite of -9 is 9.
x=\frac{9±\sqrt{21}}{10}
Multiply 2 times 5.
x=\frac{\sqrt{21}+9}{10}
Now solve the equation x=\frac{9±\sqrt{21}}{10} when ± is plus. Add 9 to \sqrt{21}.
x=\frac{9-\sqrt{21}}{10}
Now solve the equation x=\frac{9±\sqrt{21}}{10} when ± is minus. Subtract \sqrt{21} from 9.
x=\frac{\sqrt{21}+9}{10} x=\frac{9-\sqrt{21}}{10}
The equation is now solved.
5x^{2}-9x+3=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.
5x^{2}-9x+3-3=-3
Subtract 3 from both sides of the equation.
5x^{2}-9x=-3
Subtracting 3 from itself leaves 0.
\frac{5x^{2}-9x}{5}=-\frac{3}{5}
Divide both sides by 5.
x^{2}-\frac{9}{5}x=-\frac{3}{5}
Dividing by 5 undoes the multiplication by 5.
x^{2}-\frac{9}{5}x+\left(-\frac{9}{10}\right)^{2}=-\frac{3}{5}+\left(-\frac{9}{10}\right)^{2}
Divide -\frac{9}{5}, the coefficient of the x term, by 2 to get -\frac{9}{10}. Then add the square of -\frac{9}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{9}{5}x+\frac{81}{100}=-\frac{3}{5}+\frac{81}{100}
Square -\frac{9}{10} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{9}{5}x+\frac{81}{100}=\frac{21}{100}
Add -\frac{3}{5} to \frac{81}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{9}{10}\right)^{2}=\frac{21}{100}
Factor x^{2}-\frac{9}{5}x+\frac{81}{100}. 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{9}{10}\right)^{2}}=\sqrt{\frac{21}{100}}
Take the square root of both sides of the equation.
x-\frac{9}{10}=\frac{\sqrt{21}}{10} x-\frac{9}{10}=-\frac{\sqrt{21}}{10}
Simplify.
x=\frac{\sqrt{21}+9}{10} x=\frac{9-\sqrt{21}}{10}
Add \frac{9}{10} to both sides of the equation.
x ^ 2 -\frac{9}{5}x +\frac{3}{5} = 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 5
r + s = \frac{9}{5} rs = \frac{3}{5}
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{9}{10} - u s = \frac{9}{10} + u
Two numbers r and s sum up to \frac{9}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{9}{5} = \frac{9}{10}. 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{9}{10} - u) (\frac{9}{10} + u) = \frac{3}{5}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{5}
\frac{81}{100} - u^2 = \frac{3}{5}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{3}{5}-\frac{81}{100} = -\frac{21}{100}
Simplify the expression by subtracting \frac{81}{100} on both sides
u^2 = \frac{21}{100} u = \pm\sqrt{\frac{21}{100}} = \pm \frac{\sqrt{21}}{10}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{9}{10} - \frac{\sqrt{21}}{10} = 0.442 s = \frac{9}{10} + \frac{\sqrt{21}}{10} = 1.358
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.