Solve for x
x = \frac{5 \sqrt{7} + 7}{9} \approx 2.247639617
x=\frac{7-5\sqrt{7}}{9}\approx -0.692084062
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9x^{2}-14x-14=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(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 9\left(-14\right)}}{2\times 9}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -14 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-14\right)±\sqrt{196-4\times 9\left(-14\right)}}{2\times 9}
Square -14.
x=\frac{-\left(-14\right)±\sqrt{196-36\left(-14\right)}}{2\times 9}
Multiply -4 times 9.
x=\frac{-\left(-14\right)±\sqrt{196+504}}{2\times 9}
Multiply -36 times -14.
x=\frac{-\left(-14\right)±\sqrt{700}}{2\times 9}
Add 196 to 504.
x=\frac{-\left(-14\right)±10\sqrt{7}}{2\times 9}
Take the square root of 700.
x=\frac{14±10\sqrt{7}}{2\times 9}
The opposite of -14 is 14.
x=\frac{14±10\sqrt{7}}{18}
Multiply 2 times 9.
x=\frac{10\sqrt{7}+14}{18}
Now solve the equation x=\frac{14±10\sqrt{7}}{18} when ± is plus. Add 14 to 10\sqrt{7}.
x=\frac{5\sqrt{7}+7}{9}
Divide 14+10\sqrt{7} by 18.
x=\frac{14-10\sqrt{7}}{18}
Now solve the equation x=\frac{14±10\sqrt{7}}{18} when ± is minus. Subtract 10\sqrt{7} from 14.
x=\frac{7-5\sqrt{7}}{9}
Divide 14-10\sqrt{7} by 18.
x=\frac{5\sqrt{7}+7}{9} x=\frac{7-5\sqrt{7}}{9}
The equation is now solved.
9x^{2}-14x-14=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-14-\left(-14\right)=-\left(-14\right)
Add 14 to both sides of the equation.
9x^{2}-14x=-\left(-14\right)
Subtracting -14 from itself leaves 0.
9x^{2}-14x=14
Subtract -14 from 0.
\frac{9x^{2}-14x}{9}=\frac{14}{9}
Divide both sides by 9.
x^{2}-\frac{14}{9}x=\frac{14}{9}
Dividing by 9 undoes the multiplication by 9.
x^{2}-\frac{14}{9}x+\left(-\frac{7}{9}\right)^{2}=\frac{14}{9}+\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{14}{9}+\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{175}{81}
Add \frac{14}{9} 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{175}{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{175}{81}}
Take the square root of both sides of the equation.
x-\frac{7}{9}=\frac{5\sqrt{7}}{9} x-\frac{7}{9}=-\frac{5\sqrt{7}}{9}
Simplify.
x=\frac{5\sqrt{7}+7}{9} x=\frac{7-5\sqrt{7}}{9}
Add \frac{7}{9} to both sides of the equation.
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