Solve x^2+4/57(x/2)div11=3

$$x ^ { 2 } + \frac { 4 } { 5 } 7 ( \frac { x } { 2 } ) \div 11 = 3$$

Solve for x

$$x = \frac{2 \sqrt{2281} - 7}{55} \approx 1.609447845$$

$$x=\frac{-2\sqrt{2281}-7}{55}\approx -1.8639933$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 11.

$$11x^{2}+\frac{4}{5}\times 7\times \frac{x}{2}=33$$

Multiply \frac{4}{5} and 7 to get \frac{28}{5}.

$$11x^{2}+\frac{28}{5}\times \frac{x}{2}=33$$

Multiply \frac{28}{5} times \frac{x}{2} by multiplying numerator times numerator and denominator times denominator.

$$11x^{2}+\frac{28x}{5\times 2}=33$$

Cancel out 2 in both numerator and denominator.

$$11x^{2}+\frac{14x}{5}=33$$

Subtract 33 from both sides.

$$11x^{2}+\frac{14x}{5}-33=0$$

Multiply both sides of the equation by 5.

$$55x^{2}+14x-165=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 55 for a, 14 for b, and -165 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 55\left(-165\right)}}{2\times 55}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 55\left(-165\right)}}{2\times 55}$$

Multiply -4 times 55.

$$x=\frac{-14±\sqrt{196-220\left(-165\right)}}{2\times 55}$$

Multiply -220 times -165.

$$x=\frac{-14±\sqrt{196+36300}}{2\times 55}$$

Add 196 to 36300.

$$x=\frac{-14±\sqrt{36496}}{2\times 55}$$

Take the square root of 36496.

$$x=\frac{-14±4\sqrt{2281}}{2\times 55}$$

Multiply 2 times 55.

$$x=\frac{-14±4\sqrt{2281}}{110}$$

Now solve the equation x=\frac{-14±4\sqrt{2281}}{110} when ± is plus. Add -14 to 4\sqrt{2281}.

$$x=\frac{4\sqrt{2281}-14}{110}$$

Divide -14+4\sqrt{2281} by 110.

$$x=\frac{2\sqrt{2281}-7}{55}$$

Now solve the equation x=\frac{-14±4\sqrt{2281}}{110} when ± is minus. Subtract 4\sqrt{2281} from -14.

$$x=\frac{-4\sqrt{2281}-14}{110}$$

Divide -14-4\sqrt{2281} by 110.

$$x=\frac{-2\sqrt{2281}-7}{55}$$

The equation is now solved.

$$x=\frac{2\sqrt{2281}-7}{55} x=\frac{-2\sqrt{2281}-7}{55}$$

Steps for Completing the Square

Multiply both sides of the equation by 11.

$$11x^{2}+\frac{4}{5}\times 7\times \frac{x}{2}=33$$

Multiply \frac{4}{5} and 7 to get \frac{28}{5}.

$$11x^{2}+\frac{28}{5}\times \frac{x}{2}=33$$

Multiply \frac{28}{5} times \frac{x}{2} by multiplying numerator times numerator and denominator times denominator.

$$11x^{2}+\frac{28x}{5\times 2}=33$$

Cancel out 2 in both numerator and denominator.

$$11x^{2}+\frac{14x}{5}=33$$

Multiply both sides of the equation by 5.

$$55x^{2}+14x=165$$

Divide both sides by 55.

$$\frac{55x^{2}+14x}{55}=\frac{165}{55}$$

Dividing by 55 undoes the multiplication by 55.

$$x^{2}+\frac{14}{55}x=\frac{165}{55}$$

Divide 165 by 55.

$$x^{2}+\frac{14}{55}x=3$$

Divide \frac{14}{55}, the coefficient of the x term, by 2 to get \frac{7}{55}. Then add the square of \frac{7}{55} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{14}{55}x+\left(\frac{7}{55}\right)^{2}=3+\left(\frac{7}{55}\right)^{2}$$

Square \frac{7}{55} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{14}{55}x+\frac{49}{3025}=3+\frac{49}{3025}$$

Add 3 to \frac{49}{3025}.

$$x^{2}+\frac{14}{55}x+\frac{49}{3025}=\frac{9124}{3025}$$

Factor x^{2}+\frac{14}{55}x+\frac{49}{3025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{55}\right)^{2}=\frac{9124}{3025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{55}\right)^{2}}=\sqrt{\frac{9124}{3025}}$$

Simplify.

$$x+\frac{7}{55}=\frac{2\sqrt{2281}}{55} x+\frac{7}{55}=-\frac{2\sqrt{2281}}{55}$$

Subtract \frac{7}{55} from both sides of the equation.

$$x=\frac{2\sqrt{2281}-7}{55} x=\frac{-2\sqrt{2281}-7}{55}$$