Solve 5x^2-6x/2=3x-10

$$\frac { 5 x ^ { 2 } - 6 x } { 2 } = 3 x - 10$$

Solve for x (complex solution)

$$x=\frac{6}{5}+\frac{8}{5}i=1.2+1.6i$$

$$x=\frac{6}{5}-\frac{8}{5}i=1.2-1.6i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$5x^{2}-6x=6x-20$$

Subtract 6x from both sides.

$$5x^{2}-6x-6x=-20$$

Combine -6x and -6x to get -12x.

$$5x^{2}-12x=-20$$

Add 20 to both sides.

$$5x^{2}-12x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -12 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 5\times 20}}{2\times 5}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 5\times 20}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-12\right)±\sqrt{144-20\times 20}}{2\times 5}$$

Multiply -20 times 20.

$$x=\frac{-\left(-12\right)±\sqrt{144-400}}{2\times 5}$$

Add 144 to -400.

$$x=\frac{-\left(-12\right)±\sqrt{-256}}{2\times 5}$$

Take the square root of -256.

$$x=\frac{-\left(-12\right)±16i}{2\times 5}$$

The opposite of -12 is 12.

$$x=\frac{12±16i}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{12±16i}{10}$$

Now solve the equation x=\frac{12±16i}{10} when ± is plus. Add 12 to 16i.

$$x=\frac{12+16i}{10}$$

Divide 12+16i by 10.

$$x=\frac{6}{5}+\frac{8}{5}i$$

Now solve the equation x=\frac{12±16i}{10} when ± is minus. Subtract 16i from 12.

$$x=\frac{12-16i}{10}$$

Divide 12-16i by 10.

$$x=\frac{6}{5}-\frac{8}{5}i$$

The equation is now solved.

$$x=\frac{6}{5}+\frac{8}{5}i x=\frac{6}{5}-\frac{8}{5}i$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$5x^{2}-6x=6x-20$$

Subtract 6x from both sides.

$$5x^{2}-6x-6x=-20$$

Combine -6x and -6x to get -12x.

$$5x^{2}-12x=-20$$

Divide both sides by 5.

$$\frac{5x^{2}-12x}{5}=-\frac{20}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{12}{5}x=-\frac{20}{5}$$

Divide -20 by 5.

$$x^{2}-\frac{12}{5}x=-4$$

Divide -\frac{12}{5}, the coefficient of the x term, by 2 to get -\frac{6}{5}. Then add the square of -\frac{6}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{12}{5}x+\left(-\frac{6}{5}\right)^{2}=-4+\left(-\frac{6}{5}\right)^{2}$$

Square -\frac{6}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{12}{5}x+\frac{36}{25}=-4+\frac{36}{25}$$

Add -4 to \frac{36}{25}.

$$x^{2}-\frac{12}{5}x+\frac{36}{25}=-\frac{64}{25}$$

Factor x^{2}-\frac{12}{5}x+\frac{36}{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}.

$$\left(x-\frac{6}{5}\right)^{2}=-\frac{64}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{6}{5}\right)^{2}}=\sqrt{-\frac{64}{25}}$$

Simplify.

$$x-\frac{6}{5}=\frac{8}{5}i x-\frac{6}{5}=-\frac{8}{5}i$$

Add \frac{6}{5} to both sides of the equation.

$$x=\frac{6}{5}+\frac{8}{5}i x=\frac{6}{5}-\frac{8}{5}i$$