Solve x^2*10div6+6=765-2x

$${ x }^{ 2 } \times 10 \div 6+6=765-2x$$

Solve for x

$$x = \frac{3 \sqrt{1266} - 3}{5} \approx 20.74853625$$

$$x=\frac{-3\sqrt{1266}-3}{5}\approx -21.94853625$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6.

$$x^{2}\times 10+36=4590-12x$$

Subtract 4590 from both sides.

$$x^{2}\times 10+36-4590=-12x$$

Subtract 4590 from 36 to get -4554.

$$x^{2}\times 10-4554=-12x$$

Add 12x to both sides.

$$x^{2}\times 10-4554+12x=0$$

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.

$$10x^{2}+12x-4554=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 12 for b, and -4554 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 10\left(-4554\right)}}{2\times 10}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 10\left(-4554\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-12±\sqrt{144-40\left(-4554\right)}}{2\times 10}$$

Multiply -40 times -4554.

$$x=\frac{-12±\sqrt{144+182160}}{2\times 10}$$

Add 144 to 182160.

$$x=\frac{-12±\sqrt{182304}}{2\times 10}$$

Take the square root of 182304.

$$x=\frac{-12±12\sqrt{1266}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-12±12\sqrt{1266}}{20}$$

Now solve the equation x=\frac{-12±12\sqrt{1266}}{20} when ± is plus. Add -12 to 12\sqrt{1266}.

$$x=\frac{12\sqrt{1266}-12}{20}$$

Divide -12+12\sqrt{1266} by 20.

$$x=\frac{3\sqrt{1266}-3}{5}$$

Now solve the equation x=\frac{-12±12\sqrt{1266}}{20} when ± is minus. Subtract 12\sqrt{1266} from -12.

$$x=\frac{-12\sqrt{1266}-12}{20}$$

Divide -12-12\sqrt{1266} by 20.

$$x=\frac{-3\sqrt{1266}-3}{5}$$

The equation is now solved.

$$x=\frac{3\sqrt{1266}-3}{5} x=\frac{-3\sqrt{1266}-3}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 6.

$$x^{2}\times 10+36=4590-12x$$

Add 12x to both sides.

$$x^{2}\times 10+36+12x=4590$$

Subtract 36 from both sides.

$$x^{2}\times 10+12x=4590-36$$

Subtract 36 from 4590 to get 4554.

$$x^{2}\times 10+12x=4554$$

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.

$$10x^{2}+12x=4554$$

Divide both sides by 10.

$$\frac{10x^{2}+12x}{10}=\frac{4554}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{12}{10}x=\frac{4554}{10}$$

Reduce the fraction \frac{12}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{6}{5}x=\frac{4554}{10}$$

Reduce the fraction \frac{4554}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{6}{5}x=\frac{2277}{5}$$

Divide \frac{6}{5}, the coefficient of the x term, by 2 to get \frac{3}{5}. Then add the square of \frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=\frac{2277}{5}+\left(\frac{3}{5}\right)^{2}$$

Square \frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{2277}{5}+\frac{9}{25}$$

Add \frac{2277}{5} to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{11394}{25}$$

Factor x^{2}+\frac{6}{5}x+\frac{9}{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}.

$$\left(x+\frac{3}{5}\right)^{2}=\frac{11394}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{5}\right)^{2}}=\sqrt{\frac{11394}{25}}$$

Simplify.

$$x+\frac{3}{5}=\frac{3\sqrt{1266}}{5} x+\frac{3}{5}=-\frac{3\sqrt{1266}}{5}$$

Subtract \frac{3}{5} from both sides of the equation.

$$x=\frac{3\sqrt{1266}-3}{5} x=\frac{-3\sqrt{1266}-3}{5}$$