Solve For $d$:$\frac{-d}{7d+50} = \frac{1}{d+8}$There May Be 1 Or 2 Solutions.$d = \square$ Or $d = \square$

Introduction

In this article, we will delve into solving a rational equation involving a variable $d$. The given equation is $\frac{-d}{7d+50} = \frac{1}{d+8}$, and we aim to find the possible values of $d$ that satisfy this equation. We will employ algebraic techniques to manipulate the equation and isolate the variable $d$. This process will involve cross-multiplication, expansion, and simplification of the resulting expressions.

Step 1: Cross-Multiply the Rational Equation

To begin solving the equation, we will cross-multiply the rational expressions on both sides. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. The resulting equation is:

$$-d(d+8) = 7d+50$$

Step 2: Expand and Simplify the Equation

Next, we will expand the left-hand side of the equation by multiplying the terms inside the parentheses. This yields:

$$-d^2 - 8d = 7d + 50$$

Step 3: Rearrange the Terms

To isolate the variable $d$, we will move all the terms involving $d$ to one side of the equation. We will also move the constant terms to the other side. The resulting equation is:

$$-d^2 - 15d - 50 = 0$$

Step 4: Solve the Quadratic Equation

The equation we obtained in the previous step is a quadratic equation in the form $ax^2 + bx + c = 0$. We can solve this equation using the quadratic formula or by factoring. In this case, we will attempt to factor the quadratic expression.

Step 5: Factor the Quadratic Expression

We will look for two numbers whose product is $-50$ and whose sum is $-15$. These numbers are $-10$ and $-5$, so we can write the quadratic expression as:

$$-(d^2 + 15d + 50) = -(d + 10)(d + 5)$$

Step 6: Solve for $d$

Now that we have factored the quadratic expression, we can set each factor equal to zero and solve for $d$. This yields two possible solutions:

$$d + 10 = 0 \Rightarrow d = -10$$

$$d + 5 = 0 \Rightarrow d = -5$$

Conclusion

In this article, we have solved the rational equation $\frac{-d}{7d+50} = \frac{1}{d+8}$ and found two possible values of $d$. The solutions are $d = -10$ and $d = -5$. These values satisfy the original equation, and we have verified them through algebraic manipulation.

Discussion

The rational equation we solved in this article is a classic example of a quadratic equation. The process of solving the equation involved cross-multiplication, expansion, and simplification of the resulting expressions. We also factored the quadratic expression to obtain the solutions.

Applications

The techniques we used to solve the rational equation have numerous applications in mathematics and other fields. For example, they can be used to solve systems of linear equations, quadratic equations, and other types of equations. Additionally, they can be used to model real-world problems involving rational expressions.

Future Work

In future work, we can explore other types of rational equations and develop techniques for solving them. We can also investigate the properties of rational expressions and their applications in various fields.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Rational Expressions" by Math Open Reference

Keywords

• Rational equation
• Quadratic equation
• Algebraic manipulation
• Cross-multiplication
• Expansion
• Simplification
• Factoring
• Quadratic formula
• Rational expressions
• Algebra
• Mathematics