Solve {left(3/2x-frac{9/3right)}^2+left(x-3right)^2}{13}=9/4

$$\frac{ { \left( \frac{ 3 }{ 2 } x- \frac{ 9 }{ 3 } \right) }^{ 2 } + { \left(x-3 \right) }^{ 2 } }{ 13 } = \frac{ 9 }{ 4 }$$

Solve for x

$$x = \frac{3 \sqrt{165} + 30}{13} \approx 5.271976749$$

$$x=\frac{30-3\sqrt{165}}{13}\approx -0.656592134$$

Steps Using the Quadratic Formula

Multiply both sides by 13.

$$\left(\frac{3}{2}x-\frac{9}{3}\right)^{2}+\left(x-3\right)^{2}=\frac{9}{4}\times 13$$

Divide 9 by 3 to get 3.

$$\left(\frac{3}{2}x-3\right)^{2}+\left(x-3\right)^{2}=\frac{9}{4}\times 13$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{2}x-3\right)^{2}.

$$\frac{9}{4}x^{2}-9x+9+\left(x-3\right)^{2}=\frac{9}{4}\times 13$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$\frac{9}{4}x^{2}-9x+9+x^{2}-6x+9=\frac{9}{4}\times 13$$

Combine \frac{9}{4}x^{2} and x^{2} to get \frac{13}{4}x^{2}.

$$\frac{13}{4}x^{2}-9x+9-6x+9=\frac{9}{4}\times 13$$

Combine -9x and -6x to get -15x.

$$\frac{13}{4}x^{2}-15x+9+9=\frac{9}{4}\times 13$$

Add 9 and 9 to get 18.

$$\frac{13}{4}x^{2}-15x+18=\frac{9}{4}\times 13$$

Multiply \frac{9}{4} and 13 to get \frac{117}{4}.

$$\frac{13}{4}x^{2}-15x+18=\frac{117}{4}$$

Subtract \frac{117}{4} from both sides.

$$\frac{13}{4}x^{2}-15x+18-\frac{117}{4}=0$$

Subtract \frac{117}{4} from 18 to get -\frac{45}{4}.

$$\frac{13}{4}x^{2}-15x-\frac{45}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{13}{4} for a, -15 for b, and -\frac{45}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times \frac{13}{4}\left(-\frac{45}{4}\right)}}{2\times \frac{13}{4}}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times \frac{13}{4}\left(-\frac{45}{4}\right)}}{2\times \frac{13}{4}}$$

Multiply -4 times \frac{13}{4}.

$$x=\frac{-\left(-15\right)±\sqrt{225-13\left(-\frac{45}{4}\right)}}{2\times \frac{13}{4}}$$

Multiply -13 times -\frac{45}{4}.

$$x=\frac{-\left(-15\right)±\sqrt{225+\frac{585}{4}}}{2\times \frac{13}{4}}$$

Add 225 to \frac{585}{4}.

$$x=\frac{-\left(-15\right)±\sqrt{\frac{1485}{4}}}{2\times \frac{13}{4}}$$

Take the square root of \frac{1485}{4}.

$$x=\frac{-\left(-15\right)±\frac{3\sqrt{165}}{2}}{2\times \frac{13}{4}}$$

The opposite of -15 is 15.

$$x=\frac{15±\frac{3\sqrt{165}}{2}}{2\times \frac{13}{4}}$$

Multiply 2 times \frac{13}{4}.

$$x=\frac{15±\frac{3\sqrt{165}}{2}}{\frac{13}{2}}$$

Now solve the equation x=\frac{15±\frac{3\sqrt{165}}{2}}{\frac{13}{2}} when ± is plus. Add 15 to \frac{3\sqrt{165}}{2}.

$$x=\frac{\frac{3\sqrt{165}}{2}+15}{\frac{13}{2}}$$

Divide 15+\frac{3\sqrt{165}}{2} by \frac{13}{2} by multiplying 15+\frac{3\sqrt{165}}{2} by the reciprocal of \frac{13}{2}.

$$x=\frac{3\sqrt{165}+30}{13}$$

Now solve the equation x=\frac{15±\frac{3\sqrt{165}}{2}}{\frac{13}{2}} when ± is minus. Subtract \frac{3\sqrt{165}}{2} from 15.

$$x=\frac{-\frac{3\sqrt{165}}{2}+15}{\frac{13}{2}}$$

Divide 15-\frac{3\sqrt{165}}{2} by \frac{13}{2} by multiplying 15-\frac{3\sqrt{165}}{2} by the reciprocal of \frac{13}{2}.

$$x=\frac{30-3\sqrt{165}}{13}$$

The equation is now solved.

$$x=\frac{3\sqrt{165}+30}{13} x=\frac{30-3\sqrt{165}}{13}$$

Steps for Completing the Square

Multiply both sides by 13.

$$\left(\frac{3}{2}x-\frac{9}{3}\right)^{2}+\left(x-3\right)^{2}=\frac{9}{4}\times 13$$

Divide 9 by 3 to get 3.

$$\left(\frac{3}{2}x-3\right)^{2}+\left(x-3\right)^{2}=\frac{9}{4}\times 13$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{2}x-3\right)^{2}.

$$\frac{9}{4}x^{2}-9x+9+\left(x-3\right)^{2}=\frac{9}{4}\times 13$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$\frac{9}{4}x^{2}-9x+9+x^{2}-6x+9=\frac{9}{4}\times 13$$

Combine \frac{9}{4}x^{2} and x^{2} to get \frac{13}{4}x^{2}.

$$\frac{13}{4}x^{2}-9x+9-6x+9=\frac{9}{4}\times 13$$

Combine -9x and -6x to get -15x.

$$\frac{13}{4}x^{2}-15x+9+9=\frac{9}{4}\times 13$$

Add 9 and 9 to get 18.

$$\frac{13}{4}x^{2}-15x+18=\frac{9}{4}\times 13$$

Multiply \frac{9}{4} and 13 to get \frac{117}{4}.

$$\frac{13}{4}x^{2}-15x+18=\frac{117}{4}$$

Subtract 18 from both sides.

$$\frac{13}{4}x^{2}-15x=\frac{117}{4}-18$$

Subtract 18 from \frac{117}{4} to get \frac{45}{4}.

$$\frac{13}{4}x^{2}-15x=\frac{45}{4}$$

Divide both sides of the equation by \frac{13}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{13}{4}x^{2}-15x}{\frac{13}{4}}=\frac{\frac{45}{4}}{\frac{13}{4}}$$

Dividing by \frac{13}{4} undoes the multiplication by \frac{13}{4}.

$$x^{2}+\left(-\frac{15}{\frac{13}{4}}\right)x=\frac{\frac{45}{4}}{\frac{13}{4}}$$

Divide -15 by \frac{13}{4} by multiplying -15 by the reciprocal of \frac{13}{4}.

$$x^{2}-\frac{60}{13}x=\frac{\frac{45}{4}}{\frac{13}{4}}$$

Divide \frac{45}{4} by \frac{13}{4} by multiplying \frac{45}{4} by the reciprocal of \frac{13}{4}.

$$x^{2}-\frac{60}{13}x=\frac{45}{13}$$

Divide -\frac{60}{13}, the coefficient of the x term, by 2 to get -\frac{30}{13}. Then add the square of -\frac{30}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{60}{13}x+\left(-\frac{30}{13}\right)^{2}=\frac{45}{13}+\left(-\frac{30}{13}\right)^{2}$$

Square -\frac{30}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{60}{13}x+\frac{900}{169}=\frac{45}{13}+\frac{900}{169}$$

Add \frac{45}{13} to \frac{900}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{60}{13}x+\frac{900}{169}=\frac{1485}{169}$$

Factor x^{2}-\frac{60}{13}x+\frac{900}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{30}{13}\right)^{2}=\frac{1485}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{30}{13}\right)^{2}}=\sqrt{\frac{1485}{169}}$$

Simplify.

$$x-\frac{30}{13}=\frac{3\sqrt{165}}{13} x-\frac{30}{13}=-\frac{3\sqrt{165}}{13}$$

Add \frac{30}{13} to both sides of the equation.

$$x=\frac{3\sqrt{165}+30}{13} x=\frac{30-3\sqrt{165}}{13}$$