Problem 17 - Algebra 2 and Trigonometry

Question

Solve and check each of the equations. \(3 x(x-10)+80=5\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Expand the Expression

First, expand the expression on the left side of the equation: \[ 3x(x - 10) + 80 = 5 \] Use the distributive property: \[ 3x^2 - 30x + 80 = 5 \]

Step 2: Simplify and Rearrange the Equation

Move all terms to one side to form a quadratic equation: \[ 3x^2 - 30x + 80 - 5 = 0 \] Simplify: \[ 3x^2 - 30x + 75 = 0 \]

Step 3: Factor the Quadratic

Factor the quadratic equation if possible. We can factor by looking for two numbers that multiply to 225 (\(3*75\)) and add to -30. Factoring gives:\[ 3(x - 5)(x - 5) = 0 \] This simplifies to: \[ 3(x - 5)^2 = 0 \]

Step 4: Solve for x

Since the equation is \(3(x - 5)^2 = 0\): Divide by 3:\[ (x - 5)^2 = 0 \] Take the square root of both sides: \[ x - 5 = 0 \] So, \(x = 5\).

Step 5: Verify the Solution

Substitute \(x = 5\) back into the original equation:\[ 3(5)(5 - 10) + 80 = 5 \] Calculate:\[ 3(5)(-5) + 80 = 5 \] \[ -75 + 80 = 5 \] \[ 5 = 5 \] The solution satisfies the original equation.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, making it easier to multiply a single term by terms inside a set of parentheses. It's like handing out each item in a basket individually to those in the group.
For instance, in the equation given, the expression is initially \( 3x(x-10) + 80 = 5 \). To clear the parentheses, we apply the distributive property, multiplying the term outside the parentheses, 3x, by each term inside the parentheses:

• First multiply: \( 3x \times x = 3x^2 \)
• Then multiply: \( 3x \times (-10) = -30x \)

Putting these together results in \( 3x^2 - 30x \). After distributing and combining all terms, we get the equation: \( 3x^2 - 30x + 80 = 5 \). This step is vital for simplifying complex expressions and setting the stage for the solving process.

Factoring Quadratics

Factoring quadratics involves breaking down a quadratic equation into simpler expressions that can be multiplied to give the original equation. It's a bit like unwrapping and then neatly folding a large sheet into smaller squares.
In the simplified equation \( 3x^2 - 30x + 75 = 0 \), we aim to factor it. Here we look for two numbers that multiply to 225 (the product of the coefficient of \( x^2 \) and the constant term, i.e., \(3\times75\)) and add up to -30 (the coefficient of \( x \)).

• In this case, both numbers are -15 and -15 because \(-15 \times -15 = 225\) and \(-15 + -15 = -30\).

Thus, the expression factors into \( 3(x - 5)(x - 5) \), or more compactly, \( 3(x - 5)^2 \). Factoring quadratics is essential for solving these equations, as it simplifies the process of finding the variable’s value.

Solving Equations Step by Step

Solving equations step by step ensures that each action taken leads you closer to finding the correct solution. Start by transforming all terms into a readable format and simplify where possible.
For our equation, once factored as \( 3(x - 5)^2 = 0 \), we simplify it by dividing by the coefficient of the factored expression, 3, leading to \( (x - 5)^2 = 0 \).
To solve for \( x \), take the square root of both sides, resulting in \( x - 5 = 0 \). This implies that \( x = 5 \). Working step by step allows us to ensure each part of the equation is handled methodically, avoiding mistakes and yielding a reliable solution.

Verifying Solutions

Verifying your solution is like double-checking your work to ensure accuracy. This step builds confidence in your final answer. Once we have \( x = 5 \), substitute it back into the original equation, \( 3x(x-10) + 80 = 5 \), to ensure the result holds true.
By substituting \( x \) with 5:

• Calculate: \( 3(5)(5-10) + 80 \)
• This simplifies to: \( 3(5)(-5) + 80 \)
• Which becomes: \(-75 + 80 = 5 \)

As both sides equal 5, the solution is correct. Verifying acts as a final check to confirm the accuracy of your calculations and understanding of the problem.