Problem 97 - Elementary Algebra

Question

Solve. $$ -2 x(x-10)(x-1)=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\), \(x = 10\), and \(x = 1\).

Step 1: Identify the Zero Product Property

The equation \[-2x(x-10)(x-1) = 0\] is a product of three terms: \(-2x\), \((x-10)\), and \((x-1)\). To solve this, we use the Zero Product Property, which states that if a product of several factors equals zero, at least one of the factors must be zero.

Step 2: Set Each Factor to Zero

Using the Zero Product Property, set each factor equal to zero:1. \(-2x = 0\)2. \(x-10 = 0\)3. \(x-1 = 0\)You will solve these equations separately to find the values of \(x\) that satisfy each condition.

Step 3: Solve First Equation

Solve the equation \(-2x = 0\):Divide both sides by \(-2\) to get \[x = 0\]This is one solution to the original equation.

Step 4: Solve Second Equation

Solve the equation \(x-10 = 0\):Add \(10\) on both sides to isolate \(x\):\[x = 10\]This is another solution to the original equation.

Step 5: Solve Third Equation

Solve the equation \(x-1 = 0\):Add \(1\) to both sides to solve for \(x\):\[x = 1\]This is the third solution to the original equation.

Key Concepts

Factoring

Factoring is breaking down a complex expression into simpler components, called "factors". This process makes equations easier to solve. In this context, factoring helps us identify individual components that, when multiplied together, give us the original expression. For example, in the equation \[-2x(x-10)(x-1)=0,\] the expression is already factored. It's made up of three factors:

• \(-2x\)
• \((x - 10)\)
• \((x - 1)\)

Each factor contributes to the whole. Factoring tells us these pieces, which can then be individually analyzed for solutions.

Solving Equations

Solving an equation means finding the value(s) of the variable that make the equation true. For the equation \[-2x(x-10)(x-1)=0,\] we have used the Zero Product Property. This property states that if the product of several factors equals zero, then at least one of the factors must be zero. Generally, this turns one equation into several manageable ones. Setting each factor to zero gives:

• \(-2x = 0\)
• \((x-10) = 0\)
• \((x-1) = 0\)

Solving these separate equations provides the solutions to the original equation. In this case, solving them gives us

• \(x = 0\)
• \(x = 10\)
• \(x = 1\)

So, the solutions to the equation are \(x = 0\), \(x = 10\), and \(x = 1\). Each represents a point where the expression equals zero.

Polynomial Equations

Polynomial equations contain polynomials, which are expressions consisting of variables raised to whole-number exponents, constants, and coefficients combined using addition, subtraction, and multiplication. An example of a polynomial equation is \(-2x(x-10)(x-1) = 0.\) Polynomials like these often appear in factored form, making it easier to apply the Zero Product Property. Each factor of the polynomial can represent individual terms, like

• \(-2x\)
• \(x - 10\)
• \(x - 1\)

These terms combine to form a single equation that equals zero, meaning we look for values of \(x\) where the total expression drops to a value of zero. Understanding polynomials helps in recognizing how these terms relate and in determining potential solutions. This process of working with polynomial equations is integral to algebra and solving for unknown variables.