Solve 3x+2/5-7=2xx+1/2

$$\frac { 3 x + 2 } { 5 } - 7 = 2 x \frac { x + 1 } { 2 }$$

Solve for x (complex solution)

$$x=\frac{-1+2\sqrt{41}i}{5}\approx -0.2+2.561249695i$$

$$x=\frac{-2\sqrt{41}i-1}{5}\approx -0.2-2.561249695i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10, the least common multiple of 5,2.

$$2\left(3x+2\right)-70=10x\left(x+1\right)$$

Use the distributive property to multiply 2 by 3x+2.

$$6x+4-70=10x\left(x+1\right)$$

Subtract 70 from 4 to get -66.

$$6x-66=10x\left(x+1\right)$$

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

$$6x-66=10x^{2}+10x$$

Subtract 10x^{2} from both sides.

$$6x-66-10x^{2}=10x$$

Subtract 10x from both sides.

$$6x-66-10x^{2}-10x=0$$

Combine 6x and -10x to get -4x.

$$-4x-66-10x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-10x^{2}-4x-66=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, -4 for b, and -66 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-10\right)\left(-66\right)}}{2\left(-10\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-10\right)\left(-66\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-\left(-4\right)±\sqrt{16+40\left(-66\right)}}{2\left(-10\right)}$$

Multiply 40 times -66.

$$x=\frac{-\left(-4\right)±\sqrt{16-2640}}{2\left(-10\right)}$$

Add 16 to -2640.

$$x=\frac{-\left(-4\right)±\sqrt{-2624}}{2\left(-10\right)}$$

Take the square root of -2624.

$$x=\frac{-\left(-4\right)±8\sqrt{41}i}{2\left(-10\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±8\sqrt{41}i}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{4±8\sqrt{41}i}{-20}$$

Now solve the equation x=\frac{4±8\sqrt{41}i}{-20} when ± is plus. Add 4 to 8i\sqrt{41}.

$$x=\frac{4+8\sqrt{41}i}{-20}$$

Divide 4+8i\sqrt{41} by -20.

$$x=\frac{-2\sqrt{41}i-1}{5}$$

Now solve the equation x=\frac{4±8\sqrt{41}i}{-20} when ± is minus. Subtract 8i\sqrt{41} from 4.

$$x=\frac{-8\sqrt{41}i+4}{-20}$$

Divide 4-8i\sqrt{41} by -20.

$$x=\frac{-1+2\sqrt{41}i}{5}$$

The equation is now solved.

$$x=\frac{-2\sqrt{41}i-1}{5} x=\frac{-1+2\sqrt{41}i}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 10, the least common multiple of 5,2.

$$2\left(3x+2\right)-70=10x\left(x+1\right)$$

Use the distributive property to multiply 2 by 3x+2.

$$6x+4-70=10x\left(x+1\right)$$

Subtract 70 from 4 to get -66.

$$6x-66=10x\left(x+1\right)$$

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

$$6x-66=10x^{2}+10x$$

Subtract 10x^{2} from both sides.

$$6x-66-10x^{2}=10x$$

Subtract 10x from both sides.

$$6x-66-10x^{2}-10x=0$$

Combine 6x and -10x to get -4x.

$$-4x-66-10x^{2}=0$$

Add 66 to both sides. Anything plus zero gives itself.

$$-4x-10x^{2}=66$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-10x^{2}-4x=66$$

Divide both sides by -10.

$$\frac{-10x^{2}-4x}{-10}=\frac{66}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\left(-\frac{4}{-10}\right)x=\frac{66}{-10}$$

Reduce the fraction \frac{-4}{-10} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{2}{5}x=\frac{66}{-10}$$

Reduce the fraction \frac{66}{-10} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{2}{5}x=-\frac{33}{5}$$

Divide \frac{2}{5}, the coefficient of the x term, by 2 to get \frac{1}{5}. Then add the square of \frac{1}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{5}x+\left(\frac{1}{5}\right)^{2}=-\frac{33}{5}+\left(\frac{1}{5}\right)^{2}$$

Square \frac{1}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{5}x+\frac{1}{25}=-\frac{33}{5}+\frac{1}{25}$$

Add -\frac{33}{5} to \frac{1}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{5}x+\frac{1}{25}=-\frac{164}{25}$$

Factor x^{2}+\frac{2}{5}x+\frac{1}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{5}\right)^{2}=-\frac{164}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{5}\right)^{2}}=\sqrt{-\frac{164}{25}}$$

Simplify.

$$x+\frac{1}{5}=\frac{2\sqrt{41}i}{5} x+\frac{1}{5}=-\frac{2\sqrt{41}i}{5}$$

Subtract \frac{1}{5} from both sides of the equation.

$$x=\frac{-1+2\sqrt{41}i}{5} x=\frac{-2\sqrt{41}i-1}{5}$$