Solve for x
x=\frac{\sqrt{89}}{6}+\frac{1}{2}\approx 2.072330189
x=-\frac{\sqrt{89}}{6}+\frac{1}{2}\approx -1.072330189
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4x\times 9\left(x-1\right)=80
Multiply both sides of the equation by 8.
36x\left(x-1\right)=80
Multiply 4 and 9 to get 36.
36x^{2}-36x=80
Use the distributive property to multiply 36x by x-1.
36x^{2}-36x-80=0
Subtract 80 from both sides.
x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 36\left(-80\right)}}{2\times 36}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -36 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-36\right)±\sqrt{1296-4\times 36\left(-80\right)}}{2\times 36}
Square -36.
x=\frac{-\left(-36\right)±\sqrt{1296-144\left(-80\right)}}{2\times 36}
Multiply -4 times 36.
x=\frac{-\left(-36\right)±\sqrt{1296+11520}}{2\times 36}
Multiply -144 times -80.
x=\frac{-\left(-36\right)±\sqrt{12816}}{2\times 36}
Add 1296 to 11520.
x=\frac{-\left(-36\right)±12\sqrt{89}}{2\times 36}
Take the square root of 12816.
x=\frac{36±12\sqrt{89}}{2\times 36}
The opposite of -36 is 36.
x=\frac{36±12\sqrt{89}}{72}
Multiply 2 times 36.
x=\frac{12\sqrt{89}+36}{72}
Now solve the equation x=\frac{36±12\sqrt{89}}{72} when ± is plus. Add 36 to 12\sqrt{89}.
x=\frac{\sqrt{89}}{6}+\frac{1}{2}
Divide 36+12\sqrt{89} by 72.
x=\frac{36-12\sqrt{89}}{72}
Now solve the equation x=\frac{36±12\sqrt{89}}{72} when ± is minus. Subtract 12\sqrt{89} from 36.
x=-\frac{\sqrt{89}}{6}+\frac{1}{2}
Divide 36-12\sqrt{89} by 72.
x=\frac{\sqrt{89}}{6}+\frac{1}{2} x=-\frac{\sqrt{89}}{6}+\frac{1}{2}
The equation is now solved.
4x\times 9\left(x-1\right)=80
Multiply both sides of the equation by 8.
36x\left(x-1\right)=80
Multiply 4 and 9 to get 36.
36x^{2}-36x=80
Use the distributive property to multiply 36x by x-1.
\frac{36x^{2}-36x}{36}=\frac{80}{36}
Divide both sides by 36.
x^{2}+\left(-\frac{36}{36}\right)x=\frac{80}{36}
Dividing by 36 undoes the multiplication by 36.
x^{2}-x=\frac{80}{36}
Divide -36 by 36.
x^{2}-x=\frac{20}{9}
Reduce the fraction \frac{80}{36} to lowest terms by extracting and canceling out 4.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{20}{9}+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=\frac{20}{9}+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=\frac{89}{36}
Add \frac{20}{9} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{1}{2}\right)^{2}=\frac{89}{36}
Factor x^{2}-x+\frac{1}{4}. 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}{2}\right)^{2}}=\sqrt{\frac{89}{36}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{89}}{6} x-\frac{1}{2}=-\frac{\sqrt{89}}{6}
Simplify.
x=\frac{\sqrt{89}}{6}+\frac{1}{2} x=-\frac{\sqrt{89}}{6}+\frac{1}{2}
Add \frac{1}{2} to both sides of the equation.
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Simultaneous equation
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Limits
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