Solve xleft(2x-1right)+3^/5=3x^2-x/5+1/15

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

Solve for x (complex solution)

$$x=\frac{2\sqrt{33}i}{21}+\frac{2}{7}\approx 0.285714286+0.547101204i$$

$$x=-\frac{2\sqrt{33}i}{21}+\frac{2}{7}\approx 0.285714286-0.547101204i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 15, the least common multiple of 5,15.

$$15x\left(2x-1\right)+3\times 3^{1}=3\left(3x^{2}-x\right)+1$$

To multiply powers of the same base, add their exponents. Add 1 and 1 to get 2.

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

Use the distributive property to multiply 15x by 2x-1.

$$30x^{2}-15x+3^{2}=3\left(3x^{2}-x\right)+1$$

Calculate 3 to the power of 2 and get 9.

$$30x^{2}-15x+9=3\left(3x^{2}-x\right)+1$$

Use the distributive property to multiply 3 by 3x^{2}-x.

$$30x^{2}-15x+9=9x^{2}-3x+1$$

Subtract 9x^{2} from both sides.

$$30x^{2}-15x+9-9x^{2}=-3x+1$$

Combine 30x^{2} and -9x^{2} to get 21x^{2}.

$$21x^{2}-15x+9=-3x+1$$

Add 3x to both sides.

$$21x^{2}-15x+9+3x=1$$

Combine -15x and 3x to get -12x.

$$21x^{2}-12x+9=1$$

Subtract 1 from both sides.

$$21x^{2}-12x+9-1=0$$

Subtract 1 from 9 to get 8.

$$21x^{2}-12x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, -12 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 21\times 8}}{2\times 21}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 21\times 8}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-\left(-12\right)±\sqrt{144-84\times 8}}{2\times 21}$$

Multiply -84 times 8.

$$x=\frac{-\left(-12\right)±\sqrt{144-672}}{2\times 21}$$

Add 144 to -672.

$$x=\frac{-\left(-12\right)±\sqrt{-528}}{2\times 21}$$

Take the square root of -528.

$$x=\frac{-\left(-12\right)±4\sqrt{33}i}{2\times 21}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{33}i}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{12±4\sqrt{33}i}{42}$$

Now solve the equation x=\frac{12±4\sqrt{33}i}{42} when ± is plus. Add 12 to 4i\sqrt{33}.

$$x=\frac{12+4\sqrt{33}i}{42}$$

Divide 12+4i\sqrt{33} by 42.

$$x=\frac{2\sqrt{33}i}{21}+\frac{2}{7}$$

Now solve the equation x=\frac{12±4\sqrt{33}i}{42} when ± is minus. Subtract 4i\sqrt{33} from 12.

$$x=\frac{-4\sqrt{33}i+12}{42}$$

Divide 12-4i\sqrt{33} by 42.

$$x=-\frac{2\sqrt{33}i}{21}+\frac{2}{7}$$

The equation is now solved.

$$x=\frac{2\sqrt{33}i}{21}+\frac{2}{7} x=-\frac{2\sqrt{33}i}{21}+\frac{2}{7}$$

Steps for Completing the Square

Multiply both sides of the equation by 15, the least common multiple of 5,15.

$$15x\left(2x-1\right)+3\times 3^{1}=3\left(3x^{2}-x\right)+1$$

To multiply powers of the same base, add their exponents. Add 1 and 1 to get 2.

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

Use the distributive property to multiply 15x by 2x-1.

$$30x^{2}-15x+3^{2}=3\left(3x^{2}-x\right)+1$$

Calculate 3 to the power of 2 and get 9.

$$30x^{2}-15x+9=3\left(3x^{2}-x\right)+1$$

Use the distributive property to multiply 3 by 3x^{2}-x.

$$30x^{2}-15x+9=9x^{2}-3x+1$$

Subtract 9x^{2} from both sides.

$$30x^{2}-15x+9-9x^{2}=-3x+1$$

Combine 30x^{2} and -9x^{2} to get 21x^{2}.

$$21x^{2}-15x+9=-3x+1$$

Add 3x to both sides.

$$21x^{2}-15x+9+3x=1$$

Combine -15x and 3x to get -12x.

$$21x^{2}-12x+9=1$$

Subtract 9 from both sides.

$$21x^{2}-12x=1-9$$

Subtract 9 from 1 to get -8.

$$21x^{2}-12x=-8$$

Divide both sides by 21.

$$\frac{21x^{2}-12x}{21}=-\frac{8}{21}$$

Dividing by 21 undoes the multiplication by 21.

$$x^{2}+\left(-\frac{12}{21}\right)x=-\frac{8}{21}$$

Reduce the fraction \frac{-12}{21} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{4}{7}x=-\frac{8}{21}$$

Divide -\frac{4}{7}, the coefficient of the x term, by 2 to get -\frac{2}{7}. Then add the square of -\frac{2}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{7}x+\left(-\frac{2}{7}\right)^{2}=-\frac{8}{21}+\left(-\frac{2}{7}\right)^{2}$$

Square -\frac{2}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{7}x+\frac{4}{49}=-\frac{8}{21}+\frac{4}{49}$$

Add -\frac{8}{21} to \frac{4}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{7}x+\frac{4}{49}=-\frac{44}{147}$$

Factor x^{2}-\frac{4}{7}x+\frac{4}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{7}\right)^{2}=-\frac{44}{147}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{7}\right)^{2}}=\sqrt{-\frac{44}{147}}$$

Simplify.

$$x-\frac{2}{7}=\frac{2\sqrt{33}i}{21} x-\frac{2}{7}=-\frac{2\sqrt{33}i}{21}$$

Add \frac{2}{7} to both sides of the equation.

$$x=\frac{2\sqrt{33}i}{21}+\frac{2}{7} x=-\frac{2\sqrt{33}i}{21}+\frac{2}{7}$$