Solve 9/x+3-9x+1=0

$$\frac { 9 } { x + 3 } - 9 x + 1 = 0$$

Solve for x

$$x=\frac{\sqrt{277}-13}{9}\approx 0.404812997$$

$$x=\frac{-\sqrt{277}-13}{9}\approx -3.293701886$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$9-9x\left(x+3\right)+x+3=0$$

Use the distributive property to multiply -9x by x+3.

$$9-9x^{2}-27x+x+3=0$$

Combine -27x and x to get -26x.

$$9-9x^{2}-26x+3=0$$

Add 9 and 3 to get 12.

$$12-9x^{2}-26x=0$$

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.

$$-9x^{2}-26x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -26 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-9\right)\times 12}}{2\left(-9\right)}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\left(-9\right)\times 12}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-26\right)±\sqrt{676+36\times 12}}{2\left(-9\right)}$$

Multiply 36 times 12.

$$x=\frac{-\left(-26\right)±\sqrt{676+432}}{2\left(-9\right)}$$

Add 676 to 432.

$$x=\frac{-\left(-26\right)±\sqrt{1108}}{2\left(-9\right)}$$

Take the square root of 1108.

$$x=\frac{-\left(-26\right)±2\sqrt{277}}{2\left(-9\right)}$$

The opposite of -26 is 26.

$$x=\frac{26±2\sqrt{277}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{26±2\sqrt{277}}{-18}$$

Now solve the equation x=\frac{26±2\sqrt{277}}{-18} when ± is plus. Add 26 to 2\sqrt{277}.

$$x=\frac{2\sqrt{277}+26}{-18}$$

Divide 26+2\sqrt{277} by -18.

$$x=\frac{-\sqrt{277}-13}{9}$$

Now solve the equation x=\frac{26±2\sqrt{277}}{-18} when ± is minus. Subtract 2\sqrt{277} from 26.

$$x=\frac{26-2\sqrt{277}}{-18}$$

Divide 26-2\sqrt{277} by -18.

$$x=\frac{\sqrt{277}-13}{9}$$

The equation is now solved.

$$x=\frac{-\sqrt{277}-13}{9} x=\frac{\sqrt{277}-13}{9}$$

Steps for Completing the Square

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$9-9x\left(x+3\right)+x+3=0$$

Use the distributive property to multiply -9x by x+3.

$$9-9x^{2}-27x+x+3=0$$

Combine -27x and x to get -26x.

$$9-9x^{2}-26x+3=0$$

Add 9 and 3 to get 12.

$$12-9x^{2}-26x=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$-9x^{2}-26x=-12$$

Divide both sides by -9.

$$\frac{-9x^{2}-26x}{-9}=-\frac{12}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{26}{-9}\right)x=-\frac{12}{-9}$$

Divide -26 by -9.

$$x^{2}+\frac{26}{9}x=-\frac{12}{-9}$$

Reduce the fraction \frac{-12}{-9} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{26}{9}x=\frac{4}{3}$$

Divide \frac{26}{9}, the coefficient of the x term, by 2 to get \frac{13}{9}. Then add the square of \frac{13}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{26}{9}x+\left(\frac{13}{9}\right)^{2}=\frac{4}{3}+\left(\frac{13}{9}\right)^{2}$$

Square \frac{13}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{26}{9}x+\frac{169}{81}=\frac{4}{3}+\frac{169}{81}$$

Add \frac{4}{3} to \frac{169}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{26}{9}x+\frac{169}{81}=\frac{277}{81}$$

Factor x^{2}+\frac{26}{9}x+\frac{169}{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}.

$$\left(x+\frac{13}{9}\right)^{2}=\frac{277}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{9}\right)^{2}}=\sqrt{\frac{277}{81}}$$

Simplify.

$$x+\frac{13}{9}=\frac{\sqrt{277}}{9} x+\frac{13}{9}=-\frac{\sqrt{277}}{9}$$

Subtract \frac{13}{9} from both sides of the equation.

$$x=\frac{\sqrt{277}-13}{9} x=\frac{-\sqrt{277}-13}{9}$$