Question - Solving Quadratic Equation with Fractions
Solution:
To solve the equation $$ x + 2 - \frac{15}{x} = 0 $$, we can start by finding a common denominator to combine the terms on the left-hand side of the equation. Since "x" is in the denominator of one of the terms, we will use "x" as the common denominator. Here are the steps:1. Multiply each term by "x" to eliminate the fraction.$$x(x) + 2(x) - \frac{15(x)}{x} = 0(x)$$2. Simplify the equation.$$x^2 + 2x - 15 = 0$$Now we have a quadratic equation in standard form, which we can solve either by factoring, completing the square, or using the quadratic formula. Let's try factoring first:3. Factor the quadratic.$$(x + 5)(x - 3) = 0$$4. Use the Zero Product Property which states that if a product of factors is zero, then at least one of the factors must be zero.$$x + 5 = 0 \quad \text{or} \quad x - 3 = 0$$5. Solve for "x".$$x = -5 \quad \text{or} \quad x = 3$$Therefore, the solutions to the equation are $$ x = -5 $$ and $$ x = 3 $$.
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