Solve the equation: 4/(3a) - 2/(a-1) = 3/(a-2)
Understand the Problem
The question asks to solve the given equation for the variable 'a'. This involves algebraic manipulation to isolate 'a' and find its value. We'll first rearrange the equation, simplify it, and then solve for 'a'.
Answer
$a = \frac{9 \pm \sqrt{433}}{22}$
Answer for screen readers
$a = \frac{9 + \sqrt{433}}{22}$ and $a = \frac{9 - \sqrt{433}}{22}$
Steps to Solve
- Find the Least Common Denominator (LCD)
The denominators are $3a$, $a-1$, and $a-2$. Therefore, the LCD is $3a(a-1)(a-2)$.
- Multiply both sides of the equation by the LCD
$$3a(a-1)(a-2) \left( \frac{4}{3a} - \frac{2}{a-1} \right) = 3a(a-1)(a-2) \left( \frac{3}{a-2} \right)$$
- Distribute and simplify
$$4(a-1)(a-2) - 2(3a)(a-2) = 3(3a)(a-1)$$ $$4(a^2 - 3a + 2) - 6a(a-2) = 9a(a-1)$$ $$4a^2 - 12a + 8 - 6a^2 + 12a = 9a^2 - 9a$$
- Combine like terms
$$-2a^2 + 8 = 9a^2 - 9a$$
- Move all terms to one side of the equation to set it to zero
$$0 = 11a^2 - 9a - 8$$
- Solve the quadratic equation for $a$ using the quadratic formula
The quadratic formula is given by: $$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ In this case, $a=11$, $b=-9$, and $c=-8$.
- Substitute the values into the quadratic formula
$$a = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(11)(-8)}}{2(11)}$$ $$a = \frac{9 \pm \sqrt{81 + 352}}{22}$$ $$a = \frac{9 \pm \sqrt{433}}{22}$$
- Final solutions
Thus, the solutions are: $$a = \frac{9 + \sqrt{433}}{22} \quad \text{and} \quad a = \frac{9 - \sqrt{433}}{22}$$
$a = \frac{9 + \sqrt{433}}{22}$ and $a = \frac{9 - \sqrt{433}}{22}$
More Information
The quadratic formula is a fundamental tool for solving equations of the form $ax^2 + bx + c = 0$. It provides a direct method to find the roots of any quadratic equation, regardless of whether it can be easily factored.
Tips
- Incorrectly distributing or simplifying terms: Double-check each step when multiplying and combining like terms to avoid sign errors or miscalculations. A common mistake is to not distribute the $4$, $-6a$, or $9a$ correctly in step 3.
- Errors in applying the quadratic formula: Ensure that you correctly identify the coefficients $a$, $b$, and $c$ and substitute them accurately into the formula. Pay close attention to signs.
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