Problem 38 - Elementary Algebra

Question

Choose the appropriate method to solve the following. $$ 50 y 2=-10 y $$

Step-by-Step Solution

Answer

The solutions are \( y = 0 \) and \( y = -\frac{1}{5} \).

Step 1: Rearrange the Equation

The equation given is \( 50y^2 = -10y \). First, rearrange the equation by moving all terms to one side to get: \( 50y^2 + 10y = 0 \). This equation is now set to zero, which is a good start for solving quadratic equations.

Step 2: Factor the Quadratic Equation

Look for common factors in the equation \( 50y^2 + 10y = 0 \). You can factor out the greatest common factor, which is \( 10y \). The equation factored becomes: \( 10y(5y + 1) = 0 \).

Step 3: Set Each Factor to Zero

Next, use the Zero Product Property, which states if a product of two factors is zero, at least one of the factors must be zero. Set each factor of the equation \( 10y(5y + 1) = 0 \) to zero: 1. \( 10y = 0 \) 2. \( 5y + 1 = 0 \).

Step 4: Solve Each Equation

Solve each of the equations from the previous step:1. For \( 10y = 0 \), divide both sides by 10 to get \( y = 0 \).2. For \( 5y + 1 = 0 \), subtract 1 from both sides to get \( 5y = -1 \). Then divide both sides by 5, giving \( y = -\frac{1}{5} \).

Key Concepts

Factoring

One of the key techniques in solving quadratic equations is factoring. Factoring means finding the expression that, when multiplied together, give you the original equation. This process helps in simplifying the problem.
In the context of solving quadratics, the common factors are either numbers or variables that are shared by each term in the equation. In our example, the expression is \( 50y^2 + 10y \). We identify the greatest common factor, which is \( 10y \), because it divides both terms.

• The term \( 50y^2 \) simplifies to \( 10y \times 5y \).
• The term \( 10y \) simplifies to \( 10y \times 1 \).

This way, the factored equation becomes \( 10y(5y + 1) = 0 \). Factoring plays an essential role in breaking down the quadratic equation into simpler components.

Zero Product Property

Once a quadratic equation has been successfully factored, the next step usually involves the Zero Product Property. This property allows us to solve the factored equation with much ease.
According to the Zero Product Property, if the product of two factors equals zero, then at least one of the factors must be zero. This principle is very helpful because if we have an equation like \( 10y(5y + 1) = 0 \), we can simply set each factor to zero. This means:

• \( 10y = 0 \)
• \( 5y + 1 = 0 \)

This step effectively breaks the problem down into smaller, more manageable equations that are easier to solve. It is a logical and straightforward approach to solving quadratic equations.

Solving Quadratic Equations

Solving quadratic equations typically involves finding values of the variable that make the equation true. With our factors set to zero, we translate them into simple equations. For \( 10y = 0 \), dividing both sides by 10 gives us \( y = 0 \).
For the equation \( 5y + 1 = 0 \), we perform the following calculations:

• Subtract 1 from both sides: \( 5y = -1 \).
• Divide both sides by 5: \( y = -\frac{1}{5} \).

Thus, the solutions to the original quadratic equation are \( y = 0 \) and \( y = -\frac{1}{5} \). These solutions represent the values at which the original equation holds true. Every step uses simple arithmetic operations, making the process approachable for anyone learning to solve quadratics.

Rearranging Equations

Rearranging equations is often the first crucial step when solving quadratics. Initially, the given equation was \( 50y^2 = -10y \). To solve it, all terms need to be on one side to obtain the standard quadratic form \( ax^2 + bx + c = 0 \).
We achieve this by moving all terms over to one side, resulting in \( 50y^2 + 10y = 0 \). Rearranging ensures that the equation is set to zero, a necessary condition for applying other methods like factoring and the Zero Product Property. It simplifies equations and sets a solid foundation for further solution steps.