Rational Equation Solver
Algebra
Intro: Clear denominators carefully and verify domain restrictions.
Worked example
- Solve $\displaystyle \frac{2}{x-1}+\frac{3}{x+2}=1$
- Goal: Solve the rational equation $\dfrac{2}{x-1}+\dfrac{3}{x+2}=1$.
- Step 1 — Domain restrictions: Denominators cannot be zero. So require $x-1\ne0$ and $x+2\ne0$. Excluded values: $\boxed{x\ne1,\;-2}$.
- Step 2 — Identify a common denominator (LCD): $(x-1)(x+2)$.
- Step 3 — Multiply both sides by the LCD to clear fractions:
- $$ (x-1)(x+2)\left[\frac{2}{x-1}+\frac{3}{x+2}\right] = (x-1)(x+2)\cdot1. $$
- Step 4 — Cancel factors term-by-term:
- • With $\dfrac{2}{x-1}$, the $(x-1)$ cancels leaving $2(x+2)$.
- • With $\dfrac{3}{x+2}$, the $(x+2)$ cancels leaving $3(x-1)$.
- • Right side is $(x-1)(x+2)$ unchanged.
- So we get: $$2(x+2) + 3(x-1) = (x-1)(x+2).$$
- Step 5 — Expand both sides:
- • Left: $2x+4 + 3x-3 = 5x+1.$
- • Right (FOIL): $(x-1)(x+2)=x^2+x-2.$
- Thus: $$5x+1 = x^2 + x - 2.$$
- Step 6 — Bring all terms to one side (standard quadratic form):
- $$0 = x^2 + x - 2 - (5x+1) = x^2 - 4x - 3.$$
- Step 7 — Solve the quadratic $x^2 - 4x - 3 = 0$ using the quadratic formula:
- Discriminant: $\Delta = (-4)^2 - 4(1)(-3) = 16 + 12 = 28.$
- $$x = \frac{4 \pm \sqrt{28}}{2} = \frac{4 \pm 2\sqrt{7}}{2} = 2 \pm \sqrt{7}.$$
- Step 8 — Check against domain restrictions ($x\ne1,-2$):
- • $2+\sqrt{7} \approx 4.6458$ (allowed).
- • $2-\sqrt{7} \approx -0.6458$ (allowed).
- Both are valid (neither equals $1$ nor $-2$).
- Step 9 — (Optional numeric verification)
- • For $x=2+\sqrt{7}$: compute LHS $= \dfrac{2}{(2+\sqrt{7})-1}+\dfrac{3}{(2+\sqrt{7})+2}$ which evaluates numerically to $1$ (to rounding).
- • For $x=2-\sqrt{7}$: similarly, LHS evaluates to $1$ (to rounding).
- Final Answer: $$\boxed{\{\,2+\sqrt{7},\;2-\sqrt{7}\,\}}.$$
FAQs
Undefined points?
We exclude values that zero denominators.
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How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.