Solve one fifteenth of x squared equals one sixth of x plus one tenth.

Step-by-step solution

1. 1

   Find the least common denominator. The denominators are $15=3\cdot 5$, $6=2\cdot 3$ and $10=2\cdot 5$, so the LCD is $2\cdot 3\cdot 5=30$. Multiplying by $30$ (rather than by $15\cdot 6\cdot 10=900$) keeps the numbers small.

2. 2

   Multiply every term by 30.

   $30\cdot\frac{x^2}{15}=2x^2,\qquad 30\cdot\frac{x}{6}=5x,\qquad 30\cdot\frac{1}{10}=3$

   so the equation becomes $2x^2=5x+3$.

3. 3

   Put it in standard form.

   $2x^2-5x-3=0$

4. 4

   Factor by splitting the middle term. We need two numbers with product $2\cdot(-3)=-6$ and sum $-5$: those are $-6$ and $+1$:

   $2x^2-6x+x-3=2x(x-3)+1(x-3)=(2x+1)(x-3)=0$

5. 5

   Solve each factor and check.

   $2x+1=0\Rightarrow x=-\frac12,\qquad x-3=0\Rightarrow x=3$

   Check $x=3$: $\frac{9}{15}=0.6$ and $\frac36+\frac{1}{10}=0.5+0.1=0.6$ ✓. Check $x=-\frac12$: $\frac{1/4}{15}=\frac{1}{60}$ and $-\frac{1}{12}+\frac{1}{10}=\frac{-5+6}{60}=\frac{1}{60}$ ✓.