Solve 216/5x-3=12quadquadquad*5x-3

$$\frac { 216 } { 5 x - 3 } = 12 \quad \quad \quad \times 5 x - 3$$

Solve for x

$$x=\frac{13-\sqrt{1273}}{40}\approx -0.566978139$$

$$x = \frac{\sqrt{1273} + 13}{40} \approx 1.216978139$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{3}{5} since division by zero is not defined. Multiply both sides of the equation by 5x-3.

$$216=12\times 5x\left(5x-3\right)+\left(5x-3\right)\left(-3\right)$$

Multiply 12 and 5 to get 60.

$$216=60x\left(5x-3\right)+\left(5x-3\right)\left(-3\right)$$

Use the distributive property to multiply 60x by 5x-3.

$$216=300x^{2}-180x+\left(5x-3\right)\left(-3\right)$$

Use the distributive property to multiply 5x-3 by -3.

$$216=300x^{2}-180x-15x+9$$

Combine -180x and -15x to get -195x.

$$216=300x^{2}-195x+9$$

Swap sides so that all variable terms are on the left hand side.

$$300x^{2}-195x+9=216$$

Subtract 216 from both sides.

$$300x^{2}-195x+9-216=0$$

Subtract 216 from 9 to get -207.

$$300x^{2}-195x-207=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 300 for a, -195 for b, and -207 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-195\right)±\sqrt{\left(-195\right)^{2}-4\times 300\left(-207\right)}}{2\times 300}$$

Square -195.

$$x=\frac{-\left(-195\right)±\sqrt{38025-4\times 300\left(-207\right)}}{2\times 300}$$

Multiply -4 times 300.

$$x=\frac{-\left(-195\right)±\sqrt{38025-1200\left(-207\right)}}{2\times 300}$$

Multiply -1200 times -207.

$$x=\frac{-\left(-195\right)±\sqrt{38025+248400}}{2\times 300}$$

Add 38025 to 248400.

$$x=\frac{-\left(-195\right)±\sqrt{286425}}{2\times 300}$$

Take the square root of 286425.

$$x=\frac{-\left(-195\right)±15\sqrt{1273}}{2\times 300}$$

The opposite of -195 is 195.

$$x=\frac{195±15\sqrt{1273}}{2\times 300}$$

Multiply 2 times 300.

$$x=\frac{195±15\sqrt{1273}}{600}$$

Now solve the equation x=\frac{195±15\sqrt{1273}}{600} when ± is plus. Add 195 to 15\sqrt{1273}.

$$x=\frac{15\sqrt{1273}+195}{600}$$

Divide 195+15\sqrt{1273} by 600.

$$x=\frac{\sqrt{1273}+13}{40}$$

Now solve the equation x=\frac{195±15\sqrt{1273}}{600} when ± is minus. Subtract 15\sqrt{1273} from 195.

$$x=\frac{195-15\sqrt{1273}}{600}$$

Divide 195-15\sqrt{1273} by 600.

$$x=\frac{13-\sqrt{1273}}{40}$$

The equation is now solved.

$$x=\frac{\sqrt{1273}+13}{40} x=\frac{13-\sqrt{1273}}{40}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{3}{5} since division by zero is not defined. Multiply both sides of the equation by 5x-3.

$$216=12\times 5x\left(5x-3\right)+\left(5x-3\right)\left(-3\right)$$

Multiply 12 and 5 to get 60.

$$216=60x\left(5x-3\right)+\left(5x-3\right)\left(-3\right)$$

Use the distributive property to multiply 60x by 5x-3.

$$216=300x^{2}-180x+\left(5x-3\right)\left(-3\right)$$

Use the distributive property to multiply 5x-3 by -3.

$$216=300x^{2}-180x-15x+9$$

Combine -180x and -15x to get -195x.

$$216=300x^{2}-195x+9$$

Swap sides so that all variable terms are on the left hand side.

$$300x^{2}-195x+9=216$$

Subtract 9 from both sides.

$$300x^{2}-195x=216-9$$

Subtract 9 from 216 to get 207.

$$300x^{2}-195x=207$$

Divide both sides by 300.

$$\frac{300x^{2}-195x}{300}=\frac{207}{300}$$

Dividing by 300 undoes the multiplication by 300.

$$x^{2}+\left(-\frac{195}{300}\right)x=\frac{207}{300}$$

Reduce the fraction \frac{-195}{300} to lowest terms by extracting and canceling out 15.

$$x^{2}-\frac{13}{20}x=\frac{207}{300}$$

Reduce the fraction \frac{207}{300} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{13}{20}x=\frac{69}{100}$$

Divide -\frac{13}{20}, the coefficient of the x term, by 2 to get -\frac{13}{40}. Then add the square of -\frac{13}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{20}x+\left(-\frac{13}{40}\right)^{2}=\frac{69}{100}+\left(-\frac{13}{40}\right)^{2}$$

Square -\frac{13}{40} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{20}x+\frac{169}{1600}=\frac{69}{100}+\frac{169}{1600}$$

Add \frac{69}{100} to \frac{169}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{20}x+\frac{169}{1600}=\frac{1273}{1600}$$

Factor x^{2}-\frac{13}{20}x+\frac{169}{1600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{40}\right)^{2}=\frac{1273}{1600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{40}\right)^{2}}=\sqrt{\frac{1273}{1600}}$$

Simplify.

$$x-\frac{13}{40}=\frac{\sqrt{1273}}{40} x-\frac{13}{40}=-\frac{\sqrt{1273}}{40}$$

Add \frac{13}{40} to both sides of the equation.

$$x=\frac{\sqrt{1273}+13}{40} x=\frac{13-\sqrt{1273}}{40}$$