- Rearrange the equation into the form \(y^2 + 18y = 0\).
- Factor the expression to get \(y(y + 18) = 0\).
- Apply the zero-product property to find the roots.
- The solutions to the equation are \(\boxed{0, -18}\).
Explanation
- Analysis of the Problem
The equation provided is \(y^2 = -18y\). This is a quadratic equation where the variable is \(y\). To solve for \(y\), we need to find all values that satisfy this equality. We can rearrange the equation into a standard quadratic form by moving all terms to one side.
- Rearranging the Equation
We add \(18y\) to both sides of the equation to set it to zero:
\[y^2 + 18y = 0\]
This is a standard quadratic equation of the form \(ay^2 + by + c = 0\), where \(a=1\), \(b=18\), and \(c=0\).
- Factoring the Expression
To solve the equation \(y^2 + 18y = 0\), we look for common factors in the terms. Both \(y^2\) and \(18y\) share a common factor of \(y\). Factoring out \(y\), we get:
\[y(y + 18) = 0\]
This factorization allows us to use the zero-product property, which states that if the product of two or more factors is equal to zero, then at least one of the factors must be zero.
- Solving for y
Based on the zero-product property, we have two possible cases:
- \(y = 0\)
- \(y + 18 = 0\)
Solving the second case for \(y\), we subtract 18 from both sides to get \(y = -18\). Thus, the solutions to the equation are \(y = 0\) and \(y = -18\). Both values satisfy the original equation: \(0^2 = -18(0)\) and \((-18)^2 = 324 = -18(-18)\). The solutions are \(\boxed{0, -18}\).
Examples
Quadratic equations like \(y^2 = -18y\) are fundamental in physics and engineering. For example, if you are calculating the time it takes for an object to reach the ground after being launched, you might end up with an equation representing its vertical position. Solving for \(y\) (representing time or position) helps determine critical moments in motion, such as when the object is at a specific height or back at the start. Mastering these equations allows you to predict outcomes in dynamic systems.