Solve 3x^2/4x-1-x-1/2=x-2/5

$$\frac { 3 x ^ { 2 } } { 4 x - 1 } - \frac { x - 1 } { 2 } = \frac { x - 2 } { 5 }$$

Solve for x

$$x=\frac{\sqrt{1921}-43}{4}\approx 0.2073035$$

$$x=\frac{-\sqrt{1921}-43}{4}\approx -21.7073035$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{1}{4} since division by zero is not defined. Multiply both sides of the equation by 10\left(4x-1\right), the least common multiple of 4x-1,2,5.

$$10\times 3x^{2}-\left(20x-5\right)\left(x-1\right)=\left(8x-2\right)\left(x-2\right)$$

Multiply 10 and 3 to get 30.

$$30x^{2}-\left(20x-5\right)\left(x-1\right)=\left(8x-2\right)\left(x-2\right)$$

Use the distributive property to multiply 20x-5 by x-1 and combine like terms.

$$30x^{2}-\left(20x^{2}-25x+5\right)=\left(8x-2\right)\left(x-2\right)$$

To find the opposite of 20x^{2}-25x+5, find the opposite of each term.

$$30x^{2}-20x^{2}+25x-5=\left(8x-2\right)\left(x-2\right)$$

Combine 30x^{2} and -20x^{2} to get 10x^{2}.

$$10x^{2}+25x-5=\left(8x-2\right)\left(x-2\right)$$

Use the distributive property to multiply 8x-2 by x-2 and combine like terms.

$$10x^{2}+25x-5=8x^{2}-18x+4$$

Subtract 8x^{2} from both sides.

$$10x^{2}+25x-5-8x^{2}=-18x+4$$

Combine 10x^{2} and -8x^{2} to get 2x^{2}.

$$2x^{2}+25x-5=-18x+4$$

Add 18x to both sides.

$$2x^{2}+25x-5+18x=4$$

Combine 25x and 18x to get 43x.

$$2x^{2}+43x-5=4$$

Subtract 4 from both sides.

$$2x^{2}+43x-5-4=0$$

Subtract 4 from -5 to get -9.

$$2x^{2}+43x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 43 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-43±\sqrt{43^{2}-4\times 2\left(-9\right)}}{2\times 2}$$

Square 43.

$$x=\frac{-43±\sqrt{1849-4\times 2\left(-9\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-43±\sqrt{1849-8\left(-9\right)}}{2\times 2}$$

Multiply -8 times -9.

$$x=\frac{-43±\sqrt{1849+72}}{2\times 2}$$

Add 1849 to 72.

$$x=\frac{-43±\sqrt{1921}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-43±\sqrt{1921}}{4}$$

Now solve the equation x=\frac{-43±\sqrt{1921}}{4} when ± is plus. Add -43 to \sqrt{1921}.

$$x=\frac{\sqrt{1921}-43}{4}$$

Now solve the equation x=\frac{-43±\sqrt{1921}}{4} when ± is minus. Subtract \sqrt{1921} from -43.

$$x=\frac{-\sqrt{1921}-43}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{1921}-43}{4} x=\frac{-\sqrt{1921}-43}{4}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{1}{4} since division by zero is not defined. Multiply both sides of the equation by 10\left(4x-1\right), the least common multiple of 4x-1,2,5.

$$10\times 3x^{2}-\left(20x-5\right)\left(x-1\right)=\left(8x-2\right)\left(x-2\right)$$

Multiply 10 and 3 to get 30.

$$30x^{2}-\left(20x-5\right)\left(x-1\right)=\left(8x-2\right)\left(x-2\right)$$

Use the distributive property to multiply 20x-5 by x-1 and combine like terms.

$$30x^{2}-\left(20x^{2}-25x+5\right)=\left(8x-2\right)\left(x-2\right)$$

To find the opposite of 20x^{2}-25x+5, find the opposite of each term.

$$30x^{2}-20x^{2}+25x-5=\left(8x-2\right)\left(x-2\right)$$

Combine 30x^{2} and -20x^{2} to get 10x^{2}.

$$10x^{2}+25x-5=\left(8x-2\right)\left(x-2\right)$$

Use the distributive property to multiply 8x-2 by x-2 and combine like terms.

$$10x^{2}+25x-5=8x^{2}-18x+4$$

Subtract 8x^{2} from both sides.

$$10x^{2}+25x-5-8x^{2}=-18x+4$$

Combine 10x^{2} and -8x^{2} to get 2x^{2}.

$$2x^{2}+25x-5=-18x+4$$

Add 18x to both sides.

$$2x^{2}+25x-5+18x=4$$

Combine 25x and 18x to get 43x.

$$2x^{2}+43x-5=4$$

Add 5 to both sides.

$$2x^{2}+43x=4+5$$

Add 4 and 5 to get 9.

$$2x^{2}+43x=9$$

Divide both sides by 2.

$$\frac{2x^{2}+43x}{2}=\frac{9}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{43}{2}x=\frac{9}{2}$$

Divide \frac{43}{2}, the coefficient of the x term, by 2 to get \frac{43}{4}. Then add the square of \frac{43}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{43}{2}x+\left(\frac{43}{4}\right)^{2}=\frac{9}{2}+\left(\frac{43}{4}\right)^{2}$$

Square \frac{43}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{43}{2}x+\frac{1849}{16}=\frac{9}{2}+\frac{1849}{16}$$

Add \frac{9}{2} to \frac{1849}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{43}{2}x+\frac{1849}{16}=\frac{1921}{16}$$

Factor x^{2}+\frac{43}{2}x+\frac{1849}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{43}{4}\right)^{2}=\frac{1921}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{43}{4}\right)^{2}}=\sqrt{\frac{1921}{16}}$$

Simplify.

$$x+\frac{43}{4}=\frac{\sqrt{1921}}{4} x+\frac{43}{4}=-\frac{\sqrt{1921}}{4}$$

Subtract \frac{43}{4} from both sides of the equation.

$$x=\frac{\sqrt{1921}-43}{4} x=\frac{-\sqrt{1921}-43}{4}$$