Question

Solve each equation in Exercises 83–108 by the method of your choice. 3/(x - 3) + 5/(x - 4) = (x2 - 20)/(x2 - 7x + 12)

Accepted Answer

Step 1: Analyze the equation and identify the denominators. The denominators are (x - 3), (x - 4), and $$(x^2 - 7x + 12). $$Factorize the quadratic denominator $$x^2 - 7x + 12 $$into (x - 3)(x - 4). This simplifies the equation to: 3/(x - 3) + 5/(x - 4) = $$(x^2 - 20)/((x - 3)(x - 4)). $$Step 2: Determine the least common denominator (LCD) for all terms. The LCD is (x - 3)(x - 4). Rewrite each term in the equation with this common denominator. Step 3: Rewrite the left-hand side of the equation. Multiply the numerator and denominator of 3/(x - 3) by (x - 4), and multiply the numerator and denominator of 5/(x - 4) by (x - 3). This gives: (3(x - 4) + 5(x - 3))/((x - 3)(x - 4)). Step 4: Combine the left-hand side into a single fraction. Expand the numerators: 3(x - 4) becomes 3x - 12, and 5(x - 3) becomes 5x - 15. Add these together to get (3x - 12 + 5x - 15)/((x - 3)(x - 4)) = (8x - 27)/((x - 3)(x - 4)). Step 5: Set the left-hand side equal to the right-hand side. Now the equation is: (8x - 27)/((x - 3)(x - 4)) = $$(x^2 - 20)/((x - 3)(x - 4)). $$Since the denominators are the same, equate the numerators: 8x - 27 = $$x^2 - 20. $$Rearrange this into a standard quadratic equation: $$x^2 - 8x + 7 = 0. $$Solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Solve each equation in Exercises 83–108 by the method of your choice. 3/(x - 3) + 5/(x - 4) = (x² - 20)/(x² - 7x + 12)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Rational Expressions

Finding Common Denominators

Factoring Polynomials