Solve 4x-3/2x+1-102x-1/4x-3=3

$$\frac{ 4x-3 }{ 2x+1 } -10 \frac{ 2x-1 }{ 4x-3 } =3$$

Solve for x

$$x=\frac{5\sqrt{57}}{48}-\frac{3}{16}\approx 0.598941087$$

$$x=-\frac{5\sqrt{57}}{48}-\frac{3}{16}\approx -0.973941087$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{1}{2},\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by \left(4x-3\right)\left(2x+1\right), the least common multiple of 2x+1,4x-3.

$$\left(4x-3\right)\left(4x-3\right)-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

Multiply 4x-3 and 4x-3 to get \left(4x-3\right)^{2}.

$$\left(4x-3\right)^{2}-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-3\right)^{2}.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

Use the distributive property to multiply 3 by 4x-3.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=\left(12x-9\right)\left(2x+1\right)$$

Use the distributive property to multiply 12x-9 by 2x+1 and combine like terms.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=24x^{2}-6x-9$$

Subtract 24x^{2} from both sides.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}=-6x-9$$

Add 6x to both sides.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}+6x=-9$$

Add 9 to both sides.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}+6x+9=0$$

Use the distributive property to multiply -10 by 2x+1.

$$16x^{2}-24x+9+\left(-20x-10\right)\left(2x-1\right)-24x^{2}+6x+9=0$$

Use the distributive property to multiply -20x-10 by 2x-1 and combine like terms.

$$16x^{2}-24x+9-40x^{2}+10-24x^{2}+6x+9=0$$

Combine 16x^{2} and -40x^{2} to get -24x^{2}.

$$-24x^{2}-24x+9+10-24x^{2}+6x+9=0$$

Add 9 and 10 to get 19.

$$-24x^{2}-24x+19-24x^{2}+6x+9=0$$

Combine -24x^{2} and -24x^{2} to get -48x^{2}.

$$-48x^{2}-24x+19+6x+9=0$$

Combine -24x and 6x to get -18x.

$$-48x^{2}-18x+19+9=0$$

Add 19 and 9 to get 28.

$$-48x^{2}-18x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -48 for a, -18 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-48\right)\times 28}}{2\left(-48\right)}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-48\right)\times 28}}{2\left(-48\right)}$$

Multiply -4 times -48.

$$x=\frac{-\left(-18\right)±\sqrt{324+192\times 28}}{2\left(-48\right)}$$

Multiply 192 times 28.

$$x=\frac{-\left(-18\right)±\sqrt{324+5376}}{2\left(-48\right)}$$

Add 324 to 5376.

$$x=\frac{-\left(-18\right)±\sqrt{5700}}{2\left(-48\right)}$$

Take the square root of 5700.

$$x=\frac{-\left(-18\right)±10\sqrt{57}}{2\left(-48\right)}$$

The opposite of -18 is 18.

$$x=\frac{18±10\sqrt{57}}{2\left(-48\right)}$$

Multiply 2 times -48.

$$x=\frac{18±10\sqrt{57}}{-96}$$

Now solve the equation x=\frac{18±10\sqrt{57}}{-96} when ± is plus. Add 18 to 10\sqrt{57}.

$$x=\frac{10\sqrt{57}+18}{-96}$$

Divide 18+10\sqrt{57} by -96.

$$x=-\frac{5\sqrt{57}}{48}-\frac{3}{16}$$

Now solve the equation x=\frac{18±10\sqrt{57}}{-96} when ± is minus. Subtract 10\sqrt{57} from 18.

$$x=\frac{18-10\sqrt{57}}{-96}$$

Divide 18-10\sqrt{57} by -96.

$$x=\frac{5\sqrt{57}}{48}-\frac{3}{16}$$

The equation is now solved.

$$x=-\frac{5\sqrt{57}}{48}-\frac{3}{16} x=\frac{5\sqrt{57}}{48}-\frac{3}{16}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{1}{2},\frac{3}{4} since division by zero is not defined. Multiply both sides of the equation by \left(4x-3\right)\left(2x+1\right), the least common multiple of 2x+1,4x-3.

$$\left(4x-3\right)\left(4x-3\right)-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

Multiply 4x-3 and 4x-3 to get \left(4x-3\right)^{2}.

$$\left(4x-3\right)^{2}-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-3\right)^{2}.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

Use the distributive property to multiply 3 by 4x-3.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=\left(12x-9\right)\left(2x+1\right)$$

Use the distributive property to multiply 12x-9 by 2x+1 and combine like terms.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=24x^{2}-6x-9$$

Subtract 24x^{2} from both sides.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}=-6x-9$$

Add 6x to both sides.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}+6x=-9$$

Use the distributive property to multiply -10 by 2x+1.

$$16x^{2}-24x+9+\left(-20x-10\right)\left(2x-1\right)-24x^{2}+6x=-9$$

Use the distributive property to multiply -20x-10 by 2x-1 and combine like terms.

$$16x^{2}-24x+9-40x^{2}+10-24x^{2}+6x=-9$$

Combine 16x^{2} and -40x^{2} to get -24x^{2}.

$$-24x^{2}-24x+9+10-24x^{2}+6x=-9$$

Add 9 and 10 to get 19.

$$-24x^{2}-24x+19-24x^{2}+6x=-9$$

Combine -24x^{2} and -24x^{2} to get -48x^{2}.

$$-48x^{2}-24x+19+6x=-9$$

Combine -24x and 6x to get -18x.

$$-48x^{2}-18x+19=-9$$

Subtract 19 from both sides.

$$-48x^{2}-18x=-9-19$$

Subtract 19 from -9 to get -28.

$$-48x^{2}-18x=-28$$

Divide both sides by -48.

$$\frac{-48x^{2}-18x}{-48}=-\frac{28}{-48}$$

Dividing by -48 undoes the multiplication by -48.

$$x^{2}+\left(-\frac{18}{-48}\right)x=-\frac{28}{-48}$$

Reduce the fraction \frac{-18}{-48} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{3}{8}x=-\frac{28}{-48}$$

Reduce the fraction \frac{-28}{-48} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{3}{8}x=\frac{7}{12}$$

Divide \frac{3}{8}, the coefficient of the x term, by 2 to get \frac{3}{16}. Then add the square of \frac{3}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{8}x+\left(\frac{3}{16}\right)^{2}=\frac{7}{12}+\left(\frac{3}{16}\right)^{2}$$

Square \frac{3}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{8}x+\frac{9}{256}=\frac{7}{12}+\frac{9}{256}$$

Add \frac{7}{12} to \frac{9}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{8}x+\frac{9}{256}=\frac{475}{768}$$

Factor x^{2}+\frac{3}{8}x+\frac{9}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{16}\right)^{2}=\frac{475}{768}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{16}\right)^{2}}=\sqrt{\frac{475}{768}}$$

Simplify.

$$x+\frac{3}{16}=\frac{5\sqrt{57}}{48} x+\frac{3}{16}=-\frac{5\sqrt{57}}{48}$$

Subtract \frac{3}{16} from both sides of the equation.

$$x=\frac{5\sqrt{57}}{48}-\frac{3}{16} x=-\frac{5\sqrt{57}}{48}-\frac{3}{16}$$