Solve 5(2x-1)+3(4x+3)1/4x-1=5

$$5(2x-1)+3(4x+3) \frac{ 1 }{ 4x-1 } =5$$

Solve for x (complex solution)

$$x=\frac{19+\sqrt{399}i}{40}\approx 0.475+0.499374609i$$

$$x=\frac{-\sqrt{399}i+19}{40}\approx 0.475-0.499374609i$$

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 4x-1.

$$5\left(2x-1\right)\left(4x-1\right)+3\left(4x+3\right)\times 1=5\left(4x-1\right)$$

Use the distributive property to multiply 5 by 2x-1.

$$\left(10x-5\right)\left(4x-1\right)+3\left(4x+3\right)\times 1=5\left(4x-1\right)$$

Use the distributive property to multiply 10x-5 by 4x-1 and combine like terms.

$$40x^{2}-30x+5+3\left(4x+3\right)\times 1=5\left(4x-1\right)$$

Multiply 3 and 1 to get 3.

$$40x^{2}-30x+5+3\left(4x+3\right)=5\left(4x-1\right)$$

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

$$40x^{2}-30x+5+12x+9=5\left(4x-1\right)$$

Combine -30x and 12x to get -18x.

$$40x^{2}-18x+5+9=5\left(4x-1\right)$$

Add 5 and 9 to get 14.

$$40x^{2}-18x+14=5\left(4x-1\right)$$

Use the distributive property to multiply 5 by 4x-1.

$$40x^{2}-18x+14=20x-5$$

Subtract 20x from both sides.

$$40x^{2}-18x+14-20x=-5$$

Combine -18x and -20x to get -38x.

$$40x^{2}-38x+14=-5$$

Add 5 to both sides.

$$40x^{2}-38x+14+5=0$$

Add 14 and 5 to get 19.

$$40x^{2}-38x+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 40 for a, -38 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\times 40\times 19}}{2\times 40}$$

Square -38.

$$x=\frac{-\left(-38\right)±\sqrt{1444-4\times 40\times 19}}{2\times 40}$$

Multiply -4 times 40.

$$x=\frac{-\left(-38\right)±\sqrt{1444-160\times 19}}{2\times 40}$$

Multiply -160 times 19.

$$x=\frac{-\left(-38\right)±\sqrt{1444-3040}}{2\times 40}$$

Add 1444 to -3040.

$$x=\frac{-\left(-38\right)±\sqrt{-1596}}{2\times 40}$$

Take the square root of -1596.

$$x=\frac{-\left(-38\right)±2\sqrt{399}i}{2\times 40}$$

The opposite of -38 is 38.

$$x=\frac{38±2\sqrt{399}i}{2\times 40}$$

Multiply 2 times 40.

$$x=\frac{38±2\sqrt{399}i}{80}$$

Now solve the equation x=\frac{38±2\sqrt{399}i}{80} when ± is plus. Add 38 to 2i\sqrt{399}.

$$x=\frac{38+2\sqrt{399}i}{80}$$

Divide 38+2i\sqrt{399} by 80.

$$x=\frac{19+\sqrt{399}i}{40}$$

Now solve the equation x=\frac{38±2\sqrt{399}i}{80} when ± is minus. Subtract 2i\sqrt{399} from 38.

$$x=\frac{-2\sqrt{399}i+38}{80}$$

Divide 38-2i\sqrt{399} by 80.

$$x=\frac{-\sqrt{399}i+19}{40}$$

The equation is now solved.

$$x=\frac{19+\sqrt{399}i}{40} x=\frac{-\sqrt{399}i+19}{40}$$

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 4x-1.

$$5\left(2x-1\right)\left(4x-1\right)+3\left(4x+3\right)\times 1=5\left(4x-1\right)$$

Use the distributive property to multiply 5 by 2x-1.

$$\left(10x-5\right)\left(4x-1\right)+3\left(4x+3\right)\times 1=5\left(4x-1\right)$$

Use the distributive property to multiply 10x-5 by 4x-1 and combine like terms.

$$40x^{2}-30x+5+3\left(4x+3\right)\times 1=5\left(4x-1\right)$$

Multiply 3 and 1 to get 3.

$$40x^{2}-30x+5+3\left(4x+3\right)=5\left(4x-1\right)$$

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

$$40x^{2}-30x+5+12x+9=5\left(4x-1\right)$$

Combine -30x and 12x to get -18x.

$$40x^{2}-18x+5+9=5\left(4x-1\right)$$

Add 5 and 9 to get 14.

$$40x^{2}-18x+14=5\left(4x-1\right)$$

Use the distributive property to multiply 5 by 4x-1.

$$40x^{2}-18x+14=20x-5$$

Subtract 20x from both sides.

$$40x^{2}-18x+14-20x=-5$$

Combine -18x and -20x to get -38x.

$$40x^{2}-38x+14=-5$$

Subtract 14 from both sides.

$$40x^{2}-38x=-5-14$$

Subtract 14 from -5 to get -19.

$$40x^{2}-38x=-19$$

Divide both sides by 40.

$$\frac{40x^{2}-38x}{40}=-\frac{19}{40}$$

Dividing by 40 undoes the multiplication by 40.

$$x^{2}+\left(-\frac{38}{40}\right)x=-\frac{19}{40}$$

Reduce the fraction \frac{-38}{40} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{19}{20}x=-\frac{19}{40}$$

Divide -\frac{19}{20}, the coefficient of the x term, by 2 to get -\frac{19}{40}. Then add the square of -\frac{19}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{20}x+\left(-\frac{19}{40}\right)^{2}=-\frac{19}{40}+\left(-\frac{19}{40}\right)^{2}$$

Square -\frac{19}{40} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{20}x+\frac{361}{1600}=-\frac{19}{40}+\frac{361}{1600}$$

Add -\frac{19}{40} to \frac{361}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{20}x+\frac{361}{1600}=-\frac{399}{1600}$$

Factor x^{2}-\frac{19}{20}x+\frac{361}{1600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{40}\right)^{2}=-\frac{399}{1600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{40}\right)^{2}}=\sqrt{-\frac{399}{1600}}$$

Simplify.

$$x-\frac{19}{40}=\frac{\sqrt{399}i}{40} x-\frac{19}{40}=-\frac{\sqrt{399}i}{40}$$

Add \frac{19}{40} to both sides of the equation.

$$x=\frac{19+\sqrt{399}i}{40} x=\frac{-\sqrt{399}i+19}{40}$$