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Magazine Chapter 4: Problem 15
For Problems \(1-30\), solve each equation. $$ \frac{2 x}{x+3}-\frac{3}{x-6}=\frac{29}{x^{2}-3 x-18} $$
Short Answer
Expert verified
The solutions are \(x = 9.5\) and \(x = -2\).
Step by step solution
01
Identify the Denominator Terms
Let's first rewrite the equation given in terms of denominators:\[ \frac{2x}{x+3} - \frac{3}{x-6} = \frac{29}{x^2 - 3x - 18}\]We can see the terms in the denominator are \(x+3\), \(x-6\), and \(x^2 - 3x - 18\). Notice that \(x^2 - 3x - 18\) can be factored.
02
Factor the Quadratic Denominator
Factor \(x^2 - 3x - 18\) to find its factorization:\[x^2 - 3x - 18 = (x+3)(x-6)\]Now, the equation can be rewritten using this factorization for a common denominator.
03
Rewrite All Terms with the Common Denominator
Rewrite all terms with the common denominator \((x+3)(x-6)\):\[\frac{2x(x-6)}{(x+3)(x-6)} - \frac{3(x+3)}{(x+3)(x-6)} = \frac{29}{(x+3)(x-6)}\]
04
Multiply Through by the Common Denominator
To eliminate the denominators, multiply the entire equation by \((x+3)(x-6)\):\[2x(x-6) - 3(x+3) = 29\]This simplifies to a linear equation.
05
Expand the Equation
Expand the equation:\[2x^2 - 12x - 3x - 9 = 29\]Combine like terms to make it easier to solve.
06
Simplify and Rearrange
Combine the like terms in the equation:\[2x^2 - 15x - 9 = 29\]Subtract 29 from both sides to set the equation to zero:\[2x^2 - 15x - 38 = 0\]
07
Solve the Quadratic Equation
Use the quadratic formula to solve \(2x^2 - 15x - 38 = 0\), where \(a = 2\), \(b = -15\), and \(c = -38\).The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].Calculate the discriminant \(b^2 - 4ac = 225 + 304 = 529\) which is a perfect square.Thus, the solutions are:\[x = \frac{15 \pm \sqrt{529}}{4}\]\[x = \frac{15 \pm 23}{4}\]Solving gives two solutions: \(x = 9.5\) and \(x = -2\).
08
Verify Solutions and Consider Restrictions
Verify each solution in the original equation. Since \(x^2 - 3x - 18 = (x+3)(x-6)\), \(x eq -3\) and \(x eq 6\) to avoid division by zero. Thus both solutions are valid as they don't equal \(-3\) or \(6\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Rational Equations
A rational equation is an equation that includes at least one fraction whose numerator and denominator are polynomials. Solving these equations involves a few strategic steps:
- Identify the denominators and find a common denominator.
- Remove the fractions by multiplying through by this common denominator.
- Solve the resulting simplified equation.
- Check your solutions to ensure they don't make any denominators zero.
Factoring Quadratics
Factoring is a critical skill in intermediate algebra, especially when handling quadratic expressions. A quadratic expression typically takes the form \(ax^2 + bx + c\). Factoring is about expressing it as a product of simpler binomials.
In our exercise, the key challenge was factoring the complex quadratic \(x^2 - 3x - 18\). This particular expression factors neatly into \((x + 3)(x - 6)\). The method involves finding two numbers that multiply to give \(-18\) (the constant term) and add to give \(-3\) (the coefficient of the \(x\) term).
In our exercise, the key challenge was factoring the complex quadratic \(x^2 - 3x - 18\). This particular expression factors neatly into \((x + 3)(x - 6)\). The method involves finding two numbers that multiply to give \(-18\) (the constant term) and add to give \(-3\) (the coefficient of the \(x\) term).
- List pairs of factors of \(-18\).
- Identify which pair sums to \(-3\).
- The correct pairing, \(3\) and \(-6\), guides the factorization.
Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation. If you're faced with an expression of the form \(ax^2 + bx + c = 0\), the quadratic formula comes to the rescue:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula finds the roots of the quadratic equation by calculating the discriminant \(b^2 - 4ac\). The discriminant helps predict how many real solutions exist:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula finds the roots of the quadratic equation by calculating the discriminant \(b^2 - 4ac\). The discriminant helps predict how many real solutions exist:
- Positive discriminant: two real solutions.
- Zero discriminant: one real solution (perfect square).
- Negative discriminant: no real solutions.
Linear Equations
Linear equations are the simplest type of algebraic equations, typically expressed in the form \(ax + b = 0\). Solving linear equations involves straightforward steps:
- Isolate the variable by adding or subtracting terms from both sides.
- Divide or multiply to solve for the unknown variable.
