Problem 28 - Elementary Algebra

Question

Solve. $$ d(7-d)=0 $$

Step-by-Step Solution

Answer

d = 0 or d = 7

Step 1: Distribute Inside the Parenthesis

Distribute the term outside the parenthesis by multiplying it with each term inside the parenthesis: d(7 - d) = 0 becomes7d - d^2 = 0.

Step 2: Rearrange to Standard Quadratic Form

Rearrange the equation to fit the standard quadratic form, which is ax^2 + bx + c = 0:-d^2 + 7d = 0 or d^2 - 7d = 0.

Step 3: Factor the Quadratic Equation

Factor out the common term, which is d:d(d - 7) = 0.

Step 4: Set Each Factor to Zero

Set each factor equal to zero and solve for d:d = 0 and d - 7 = 0 Therefore, d = 0 or d = 7.

Key Concepts

quadratic equations

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \). This type of equation includes terms up to the second degree (squared). In this exercise, we deal with a quadratic equation. The equation is given as \(d(7 - d) = 0\), and we need to manipulate and solve it. Quadratic equations often have two solutions, reflecting the parabola's points where it intersects the x-axis. Let's move through the steps to understand how to solve this quadratic equation.

factoring

Factoring is the process of breaking down an equation into simpler multiples, which can then be set to zero. In the step-by-step solution, after rearranging the equation to \(d^2 - 7d = 0\), the factoring step involves finding the greatest common factor. In this case, both terms share a common factor of \(d\). So, we factor out \(d\) from the equation: \(d(d - 7) = 0\). Factoring makes it easier to solve quadratic equations because you can set each factor to zero and solve them individually.

distributive property

The distributive property is a basic property of multiplication in algebra. It states that \(a(b + c) = ab + ac\). In our problem, we use the distributive property in the first step when we go from \(d(7 - d)\) to \(7d - d^2\). This simplifies the expression by distributing \(d\) to both terms inside the parenthesis, allowing us to rearrange the equation in standard quadratic form. Understanding the distributive property helps in simplifying and ultimately solving equations.

solving equations

Solving quadratic equations involves finding the values of \(x\) (or \(d\) in this case) that make the equation true. For \(d(d - 7) = 0\), we use the zero product property, which states if \(ab = 0\), then either \(a = 0\) or \(b = 0\). This leads to solving the smaller equations \(d = 0\) and \(d - 7 = 0\). Thus, we find the solutions \(d = 0\) and \(d = 7\). Each solution represents a point where the quadratic graph touches the x-axis. Solving equations step-by-step ensures no mistakes and a clear understanding of the solution.