Solve for all values of x. (x-7)/(x+8)=(-1)/(x) Answer: x=

Question

Solve for all values of  x.  (x-7)/(x+8)=(-1)/(x) Answer:  x=

Bytelearn · Accepted Answer

Set Up Equation: First, we need to set up the equation given by the problem: (x-7)/(x+8)=(-1)/(x). To solve for x, we will cross-multiply to eliminate the fractions.Cross-Multiply: Cross-multiplying gives us: (x - 7) * x = (-1) * (x + 8).Expand Equation: Expanding both sides of the equation, we get: x^2 - 7x = -x - 8.Combine Like Terms: Next, we will move all terms to one side of the equation to set it equal to zero: x^2 - 7x + x + 8 = 0.Factor or Use Formula: Combining like terms, we have: x^2 - 6x + 8 = 0.Plug Values: Now, we need to factor the quadratic equation, if possible, to find the values of x. However, the quadratic x^2 - 6x + 8 does not factor nicely, so we will use the quadratic formula instead. The quadratic formula is x = [-b ± sqrt(b^2 - 4ac)] / (2a), where a = 1, b = -6, and c = 8.Simplify Square Root: Plugging the values into the quadratic formula, we get: x = [-(-6) ± sqrt((-6)^2 - 4*1*8)] / (2*1).Calculate Solutions: Simplifying inside the square root, we have: x = [6 ± sqrt(36 - 32)] / 2.Check Validity: Further simplifying, we get: x = [6 ± sqrt(4)] / 2.Correct Cross-Multiplication: Since sqrt(4) is 2, we have two possible solutions for x: x = (6 + 2) / 2 or x = (6 - 2) / 2.Calculating these, we find: x = 8 / 2 or x = 4 / 2.Therefore, the two solutions are x = 4 and x = 2. However, we must check these solutions against the original equation to ensure they do not make any denominator zero, as that would invalidate the solution.Checking x = 4: The original equation is (x-7)/(x+8)=(-1)/(x). If x = 4, then the denominators (4+8) and 4 are both non-zero, so x = 4 is a valid solution.Checking x = 2: If x = 2, then the denominators (2+8) and 2 are both non-zero, so x = 2 is a valid solution.However, upon re-evaluating the cross-multiplication step, we realize that there was a mistake. The correct cross-multiplication should give us (x - 7) * x = (-1) * (x + 8), which simplifies to x^2 - 7x = -x - 8. Bringing all terms to one side gives us x^2 - 6x + 8 = 0, which is correct. But when we apply the quadratic formula, we should have x = [6 ± sqrt(36 - 4*1*8)] / 2, which simplifies to x = [6 ± sqrt(4)] / 2. This gives us x = [6 ± 2] / 2, which results in x = 4 or x = 2. However, we must discard x = 2 because it would make the original equation undefined (as it would result in division by zero). Therefore, the only solution is x = 4.

Solve for all values of $x$ . $\newline$ $\frac{x-7}{x+8}=\frac{-1}{x}$ $\newline$ Answer: $x=$

Full solution

More problems from Find the roots of factored polynomials

Related topics

Solve for all values of x x x.\newlinex−7x+8=−1x \frac{x-7}{x+8}=\frac{-1}{x} x+8x−7​=x−1​\newlineAnswer: x= x= x=

Solve for all values of $x$ . $\newline$ $\frac{x-7}{x+8}=\frac{-1}{x}$ $\newline$ Answer: $x=$