Solve 5x+9/8x-1-5x+1/3x-1=1

$$\frac { 5 x + 9 } { 8 x - 1 } - \frac { 5 x + 1 } { 3 x - 1 } = 1$$

Solve for x (complex solution)

$$x=\frac{-6\sqrt{6}i+15}{49}\approx 0.306122449-0.29993752i$$

$$x=\frac{15+6\sqrt{6}i}{49}\approx 0.306122449+0.29993752i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{1}{8},\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-1\right)\left(8x-1\right), the least common multiple of 8x-1,3x-1.

$$\left(3x-1\right)\left(5x+9\right)-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

Use the distributive property to multiply 3x-1 by 5x+9 and combine like terms.

$$15x^{2}+22x-9-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

Use the distributive property to multiply 8x-1 by 5x+1 and combine like terms.

$$15x^{2}+22x-9-\left(40x^{2}+3x-1\right)=\left(3x-1\right)\left(8x-1\right)$$

To find the opposite of 40x^{2}+3x-1, find the opposite of each term.

$$15x^{2}+22x-9-40x^{2}-3x+1=\left(3x-1\right)\left(8x-1\right)$$

Combine 15x^{2} and -40x^{2} to get -25x^{2}.

$$-25x^{2}+22x-9-3x+1=\left(3x-1\right)\left(8x-1\right)$$

Combine 22x and -3x to get 19x.

$$-25x^{2}+19x-9+1=\left(3x-1\right)\left(8x-1\right)$$

Add -9 and 1 to get -8.

$$-25x^{2}+19x-8=\left(3x-1\right)\left(8x-1\right)$$

Use the distributive property to multiply 3x-1 by 8x-1 and combine like terms.

$$-25x^{2}+19x-8=24x^{2}-11x+1$$

Subtract 24x^{2} from both sides.

$$-25x^{2}+19x-8-24x^{2}=-11x+1$$

Combine -25x^{2} and -24x^{2} to get -49x^{2}.

$$-49x^{2}+19x-8=-11x+1$$

Add 11x to both sides.

$$-49x^{2}+19x-8+11x=1$$

Combine 19x and 11x to get 30x.

$$-49x^{2}+30x-8=1$$

Subtract 1 from both sides.

$$-49x^{2}+30x-8-1=0$$

Subtract 1 from -8 to get -9.

$$-49x^{2}+30x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -49 for a, 30 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-49\right)\left(-9\right)}}{2\left(-49\right)}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\left(-49\right)\left(-9\right)}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$x=\frac{-30±\sqrt{900+196\left(-9\right)}}{2\left(-49\right)}$$

Multiply 196 times -9.

$$x=\frac{-30±\sqrt{900-1764}}{2\left(-49\right)}$$

Add 900 to -1764.

$$x=\frac{-30±\sqrt{-864}}{2\left(-49\right)}$$

Take the square root of -864.

$$x=\frac{-30±12\sqrt{6}i}{2\left(-49\right)}$$

Multiply 2 times -49.

$$x=\frac{-30±12\sqrt{6}i}{-98}$$

Now solve the equation x=\frac{-30±12\sqrt{6}i}{-98} when ± is plus. Add -30 to 12i\sqrt{6}.

$$x=\frac{-30+12\sqrt{6}i}{-98}$$

Divide -30+12i\sqrt{6} by -98.

$$x=\frac{-6\sqrt{6}i+15}{49}$$

Now solve the equation x=\frac{-30±12\sqrt{6}i}{-98} when ± is minus. Subtract 12i\sqrt{6} from -30.

$$x=\frac{-12\sqrt{6}i-30}{-98}$$

Divide -30-12i\sqrt{6} by -98.

$$x=\frac{15+6\sqrt{6}i}{49}$$

The equation is now solved.

$$x=\frac{-6\sqrt{6}i+15}{49} x=\frac{15+6\sqrt{6}i}{49}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{1}{8},\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-1\right)\left(8x-1\right), the least common multiple of 8x-1,3x-1.

$$\left(3x-1\right)\left(5x+9\right)-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

Use the distributive property to multiply 3x-1 by 5x+9 and combine like terms.

$$15x^{2}+22x-9-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

Use the distributive property to multiply 8x-1 by 5x+1 and combine like terms.

$$15x^{2}+22x-9-\left(40x^{2}+3x-1\right)=\left(3x-1\right)\left(8x-1\right)$$

To find the opposite of 40x^{2}+3x-1, find the opposite of each term.

$$15x^{2}+22x-9-40x^{2}-3x+1=\left(3x-1\right)\left(8x-1\right)$$

Combine 15x^{2} and -40x^{2} to get -25x^{2}.

$$-25x^{2}+22x-9-3x+1=\left(3x-1\right)\left(8x-1\right)$$

Combine 22x and -3x to get 19x.

$$-25x^{2}+19x-9+1=\left(3x-1\right)\left(8x-1\right)$$

Add -9 and 1 to get -8.

$$-25x^{2}+19x-8=\left(3x-1\right)\left(8x-1\right)$$

Use the distributive property to multiply 3x-1 by 8x-1 and combine like terms.

$$-25x^{2}+19x-8=24x^{2}-11x+1$$

Subtract 24x^{2} from both sides.

$$-25x^{2}+19x-8-24x^{2}=-11x+1$$

Combine -25x^{2} and -24x^{2} to get -49x^{2}.

$$-49x^{2}+19x-8=-11x+1$$

Add 11x to both sides.

$$-49x^{2}+19x-8+11x=1$$

Combine 19x and 11x to get 30x.

$$-49x^{2}+30x-8=1$$

Add 8 to both sides.

$$-49x^{2}+30x=1+8$$

Add 1 and 8 to get 9.

$$-49x^{2}+30x=9$$

Divide both sides by -49.

$$\frac{-49x^{2}+30x}{-49}=\frac{9}{-49}$$

Dividing by -49 undoes the multiplication by -49.

$$x^{2}+\frac{30}{-49}x=\frac{9}{-49}$$

Divide 30 by -49.

$$x^{2}-\frac{30}{49}x=\frac{9}{-49}$$

Divide 9 by -49.

$$x^{2}-\frac{30}{49}x=-\frac{9}{49}$$

Divide -\frac{30}{49}, the coefficient of the x term, by 2 to get -\frac{15}{49}. Then add the square of -\frac{15}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{30}{49}x+\left(-\frac{15}{49}\right)^{2}=-\frac{9}{49}+\left(-\frac{15}{49}\right)^{2}$$

Square -\frac{15}{49} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{30}{49}x+\frac{225}{2401}=-\frac{9}{49}+\frac{225}{2401}$$

Add -\frac{9}{49} to \frac{225}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{30}{49}x+\frac{225}{2401}=-\frac{216}{2401}$$

Factor x^{2}-\frac{30}{49}x+\frac{225}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{49}\right)^{2}=-\frac{216}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{49}\right)^{2}}=\sqrt{-\frac{216}{2401}}$$

Simplify.

$$x-\frac{15}{49}=\frac{6\sqrt{6}i}{49} x-\frac{15}{49}=-\frac{6\sqrt{6}i}{49}$$

Add \frac{15}{49} to both sides of the equation.

$$x=\frac{15+6\sqrt{6}i}{49} x=\frac{-6\sqrt{6}i+15}{49}$$