Solve 2/5-1/3(x-15)=x^2-5

$$\frac { 2 } { 5 } - \frac { 1 } { 3 } ( x - 15 ) = x ^ { 2 } - 5$$

Solve for x

$$x=\frac{\sqrt{9385}}{30}-\frac{1}{6}\approx 3.062540322$$

$$x=-\frac{\sqrt{9385}}{30}-\frac{1}{6}\approx -3.395873656$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -\frac{1}{3} by x-15.

$$\frac{2}{5}-\frac{1}{3}x+5=x^{2}-5$$

Add \frac{2}{5} and 5 to get \frac{27}{5}.

$$\frac{27}{5}-\frac{1}{3}x=x^{2}-5$$

Subtract x^{2} from both sides.

$$\frac{27}{5}-\frac{1}{3}x-x^{2}=-5$$

Add 5 to both sides.

$$\frac{27}{5}-\frac{1}{3}x-x^{2}+5=0$$

Add \frac{27}{5} and 5 to get \frac{52}{5}.

$$\frac{52}{5}-\frac{1}{3}x-x^{2}=0$$

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.

$$-x^{2}-\frac{1}{3}x+\frac{52}{5}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{1}{3} for b, and \frac{52}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{3}\right)±\sqrt{\left(-\frac{1}{3}\right)^{2}-4\left(-1\right)\times \frac{52}{5}}}{2\left(-1\right)}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{1}{3}\right)±\sqrt{\frac{1}{9}-4\left(-1\right)\times \frac{52}{5}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{1}{3}\right)±\sqrt{\frac{1}{9}+4\times \frac{52}{5}}}{2\left(-1\right)}$$

Multiply 4 times \frac{52}{5}.

$$x=\frac{-\left(-\frac{1}{3}\right)±\sqrt{\frac{1}{9}+\frac{208}{5}}}{2\left(-1\right)}$$

Add \frac{1}{9} to \frac{208}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{1}{3}\right)±\sqrt{\frac{1877}{45}}}{2\left(-1\right)}$$

Take the square root of \frac{1877}{45}.

$$x=\frac{-\left(-\frac{1}{3}\right)±\frac{\sqrt{9385}}{15}}{2\left(-1\right)}$$

The opposite of -\frac{1}{3} is \frac{1}{3}.

$$x=\frac{\frac{1}{3}±\frac{\sqrt{9385}}{15}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{1}{3}±\frac{\sqrt{9385}}{15}}{-2}$$

Now solve the equation x=\frac{\frac{1}{3}±\frac{\sqrt{9385}}{15}}{-2} when ± is plus. Add \frac{1}{3} to \frac{\sqrt{9385}}{15}.

$$x=\frac{\frac{\sqrt{9385}}{15}+\frac{1}{3}}{-2}$$

Divide \frac{1}{3}+\frac{\sqrt{9385}}{15} by -2.

$$x=-\frac{\sqrt{9385}}{30}-\frac{1}{6}$$

Now solve the equation x=\frac{\frac{1}{3}±\frac{\sqrt{9385}}{15}}{-2} when ± is minus. Subtract \frac{\sqrt{9385}}{15} from \frac{1}{3}.

$$x=\frac{-\frac{\sqrt{9385}}{15}+\frac{1}{3}}{-2}$$

Divide \frac{1}{3}-\frac{\sqrt{9385}}{15} by -2.

$$x=\frac{\sqrt{9385}}{30}-\frac{1}{6}$$

The equation is now solved.

$$x=-\frac{\sqrt{9385}}{30}-\frac{1}{6} x=\frac{\sqrt{9385}}{30}-\frac{1}{6}$$

Steps for Completing the Square

Use the distributive property to multiply -\frac{1}{3} by x-15.

$$\frac{2}{5}-\frac{1}{3}x+5=x^{2}-5$$

Add \frac{2}{5} and 5 to get \frac{27}{5}.

$$\frac{27}{5}-\frac{1}{3}x=x^{2}-5$$

Subtract x^{2} from both sides.

$$\frac{27}{5}-\frac{1}{3}x-x^{2}=-5$$

Subtract \frac{27}{5} from both sides.

$$-\frac{1}{3}x-x^{2}=-5-\frac{27}{5}$$

Subtract \frac{27}{5} from -5 to get -\frac{52}{5}.

$$-\frac{1}{3}x-x^{2}=-\frac{52}{5}$$

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.

$$-x^{2}-\frac{1}{3}x=-\frac{52}{5}$$

Divide both sides by -1.

$$\frac{-x^{2}-\frac{1}{3}x}{-1}=-\frac{\frac{52}{5}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{\frac{1}{3}}{-1}\right)x=-\frac{\frac{52}{5}}{-1}$$

Divide -\frac{1}{3} by -1.

$$x^{2}+\frac{1}{3}x=-\frac{\frac{52}{5}}{-1}$$

Divide -\frac{52}{5} by -1.

$$x^{2}+\frac{1}{3}x=\frac{52}{5}$$

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=\frac{52}{5}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{52}{5}+\frac{1}{36}$$

Add \frac{52}{5} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{1877}{180}$$

Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=\frac{1877}{180}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{\frac{1877}{180}}$$

Simplify.

$$x+\frac{1}{6}=\frac{\sqrt{9385}}{30} x+\frac{1}{6}=-\frac{\sqrt{9385}}{30}$$

Subtract \frac{1}{6} from both sides of the equation.

$$x=\frac{\sqrt{9385}}{30}-\frac{1}{6} x=-\frac{\sqrt{9385}}{30}-\frac{1}{6}$$