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9x^{2}+14x+21=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{-14±\sqrt{14^{2}-4\times 9\times 21}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 14 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-14±\sqrt{196-4\times 9\times 21}}{2\times 9}
Square 14.
x=\frac{-14±\sqrt{196-36\times 21}}{2\times 9}
Multiply -4 times 9.
x=\frac{-14±\sqrt{196-756}}{2\times 9}
Multiply -36 times 21.
x=\frac{-14±\sqrt{-560}}{2\times 9}
Add 196 to -756.
x=\frac{-14±4\sqrt{35}i}{2\times 9}
Take the square root of -560.
x=\frac{-14±4\sqrt{35}i}{18}
Multiply 2 times 9.
x=\frac{-14+4\sqrt{35}i}{18}
Now solve the equation x=\frac{-14±4\sqrt{35}i}{18} when ± is plus. Add -14 to 4i\sqrt{35}.
x=\frac{-7+2\sqrt{35}i}{9}
Divide -14+4i\sqrt{35} by 18.
x=\frac{-4\sqrt{35}i-14}{18}
Now solve the equation x=\frac{-14±4\sqrt{35}i}{18} when ± is minus. Subtract 4i\sqrt{35} from -14.
x=\frac{-2\sqrt{35}i-7}{9}
Divide -14-4i\sqrt{35} by 18.
x=\frac{-7+2\sqrt{35}i}{9} x=\frac{-2\sqrt{35}i-7}{9}
The equation is now solved.
9x^{2}+14x+21=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}+14x+21-21=-21
Subtract 21 from both sides of the equation.
9x^{2}+14x=-21
Subtracting 21 from itself leaves 0.
\frac{9x^{2}+14x}{9}=-\frac{21}{9}
Divide both sides by 9.
x^{2}+\frac{14}{9}x=-\frac{21}{9}
Dividing by 9 undoes the multiplication by 9.
x^{2}+\frac{14}{9}x=-\frac{7}{3}
Reduce the fraction \frac{-21}{9} to lowest terms by extracting and canceling out 3.
x^{2}+\frac{14}{9}x+\left(\frac{7}{9}\right)^{2}=-\frac{7}{3}+\left(\frac{7}{9}\right)^{2}
Divide \frac{14}{9}, the coefficient of the x term, by 2 to get \frac{7}{9}. Then add the square of \frac{7}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{14}{9}x+\frac{49}{81}=-\frac{7}{3}+\frac{49}{81}
Square \frac{7}{9} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{14}{9}x+\frac{49}{81}=-\frac{140}{81}
Add -\frac{7}{3} to \frac{49}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{7}{9}\right)^{2}=-\frac{140}{81}
Factor x^{2}+\frac{14}{9}x+\frac{49}{81}. 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{7}{9}\right)^{2}}=\sqrt{-\frac{140}{81}}
Take the square root of both sides of the equation.
x+\frac{7}{9}=\frac{2\sqrt{35}i}{9} x+\frac{7}{9}=-\frac{2\sqrt{35}i}{9}
Simplify.
x=\frac{-7+2\sqrt{35}i}{9} x=\frac{-2\sqrt{35}i-7}{9}
Subtract \frac{7}{9} from both sides of the equation.
x ^ 2 +\frac{14}{9}x +\frac{7}{3} = 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{14}{9} rs = \frac{7}{3}
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{7}{9} - u s = -\frac{7}{9} + u
Two numbers r and s sum up to -\frac{14}{9} exactly when the average of the two numbers is \frac{1}{2}*-\frac{14}{9} = -\frac{7}{9}. 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{7}{9} - u) (-\frac{7}{9} + u) = \frac{7}{3}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{7}{3}
\frac{49}{81} - u^2 = \frac{7}{3}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{7}{3}-\frac{49}{81} = \frac{140}{81}
Simplify the expression by subtracting \frac{49}{81} on both sides
u^2 = -\frac{140}{81} u = \pm\sqrt{-\frac{140}{81}} = \pm \frac{\sqrt{140}}{9}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{7}{9} - \frac{\sqrt{140}}{9}i = -0.778 - 1.315i s = -\frac{7}{9} + \frac{\sqrt{140}}{9}i = -0.778 + 1.315i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.