Solve left(81xright)^2-1150x-975=0

$${ \left(81x \right) }^{ 2 } -1150x-975 = 0$$

Solve for x

$$x=\frac{220\sqrt{139}+575}{6561}\approx 0.482969326$$

$$x=\frac{575-220\sqrt{139}}{6561}\approx -0.307691167$$

Steps Using the Quadratic Formula

Expand \left(81x\right)^{2}.

$$81^{2}x^{2}-1150x-975=0$$

Calculate 81 to the power of 2 and get 6561.

$$6561x^{2}-1150x-975=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6561 for a, -1150 for b, and -975 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1150\right)±\sqrt{\left(-1150\right)^{2}-4\times 6561\left(-975\right)}}{2\times 6561}$$

Square -1150.

$$x=\frac{-\left(-1150\right)±\sqrt{1322500-4\times 6561\left(-975\right)}}{2\times 6561}$$

Multiply -4 times 6561.

$$x=\frac{-\left(-1150\right)±\sqrt{1322500-26244\left(-975\right)}}{2\times 6561}$$

Multiply -26244 times -975.

$$x=\frac{-\left(-1150\right)±\sqrt{1322500+25587900}}{2\times 6561}$$

Add 1322500 to 25587900.

$$x=\frac{-\left(-1150\right)±\sqrt{26910400}}{2\times 6561}$$

Take the square root of 26910400.

$$x=\frac{-\left(-1150\right)±440\sqrt{139}}{2\times 6561}$$

The opposite of -1150 is 1150.

$$x=\frac{1150±440\sqrt{139}}{2\times 6561}$$

Multiply 2 times 6561.

$$x=\frac{1150±440\sqrt{139}}{13122}$$

Now solve the equation x=\frac{1150±440\sqrt{139}}{13122} when ± is plus. Add 1150 to 440\sqrt{139}.

$$x=\frac{440\sqrt{139}+1150}{13122}$$

Divide 1150+440\sqrt{139} by 13122.

$$x=\frac{220\sqrt{139}+575}{6561}$$

Now solve the equation x=\frac{1150±440\sqrt{139}}{13122} when ± is minus. Subtract 440\sqrt{139} from 1150.

$$x=\frac{1150-440\sqrt{139}}{13122}$$

Divide 1150-440\sqrt{139} by 13122.

$$x=\frac{575-220\sqrt{139}}{6561}$$

The equation is now solved.

$$x=\frac{220\sqrt{139}+575}{6561} x=\frac{575-220\sqrt{139}}{6561}$$

Steps for Completing the Square

Expand \left(81x\right)^{2}.

$$81^{2}x^{2}-1150x-975=0$$

Calculate 81 to the power of 2 and get 6561.

$$6561x^{2}-1150x-975=0$$

Add 975 to both sides. Anything plus zero gives itself.

$$6561x^{2}-1150x=975$$

Divide both sides by 6561.

$$\frac{6561x^{2}-1150x}{6561}=\frac{975}{6561}$$

Dividing by 6561 undoes the multiplication by 6561.

$$x^{2}-\frac{1150}{6561}x=\frac{975}{6561}$$

Reduce the fraction \frac{975}{6561} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{1150}{6561}x=\frac{325}{2187}$$

Divide -\frac{1150}{6561}, the coefficient of the x term, by 2 to get -\frac{575}{6561}. Then add the square of -\frac{575}{6561} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1150}{6561}x+\left(-\frac{575}{6561}\right)^{2}=\frac{325}{2187}+\left(-\frac{575}{6561}\right)^{2}$$

Square -\frac{575}{6561} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1150}{6561}x+\frac{330625}{43046721}=\frac{325}{2187}+\frac{330625}{43046721}$$

Add \frac{325}{2187} to \frac{330625}{43046721} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1150}{6561}x+\frac{330625}{43046721}=\frac{6727600}{43046721}$$

Factor x^{2}-\frac{1150}{6561}x+\frac{330625}{43046721}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{575}{6561}\right)^{2}=\frac{6727600}{43046721}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{575}{6561}\right)^{2}}=\sqrt{\frac{6727600}{43046721}}$$

Simplify.

$$x-\frac{575}{6561}=\frac{220\sqrt{139}}{6561} x-\frac{575}{6561}=-\frac{220\sqrt{139}}{6561}$$

Add \frac{575}{6561} to both sides of the equation.

$$x=\frac{220\sqrt{139}+575}{6561} x=\frac{575-220\sqrt{139}}{6561}$$