Solve for w w^2=-18w
The given problem asks to find the value of the variable "w" that satisfies the equation w^2=-18w. The equation is a quadratic equation where one would typically set it to zero by moving all terms to one side and solve for "w" by factoring, completing the square, or using the quadratic formula to find the roots of the equation. The question requires determining the value(s) of "w" that make the equation true.
$w^{2} = - 18 w$
Answer
Solution:
Step 1:
Move $18w$ to the left side by adding it to both sides: $w^2 + 18w = 0$.
Step 2:
Extract the common factor $w$ from the terms $w^2$ and $18w$.
Step 2.1:
Take $w$ out of $w^2$: $w \cdot w + 18w = 0$.
Step 2.2:
Take $w$ out of $18w$: $w \cdot w + w \cdot 18 = 0$.
Step 2.3:
Combine the factored terms: $w(w + 18) = 0$.
Step 3:
Recognize that if any factor in the product $w(w + 18)$ is zero, the product is zero: $w = 0$ or $w + 18 = 0$.
Step 4:
The first solution is when $w$ itself is zero: $w = 0$.
Step 5:
Find the second solution by setting $w + 18$ equal to zero and solving for $w$.
Step 5.1:
Write down the equation: $w + 18 = 0$.
Step 5.2:
Isolate $w$ by subtracting $18$: $w = -18$.
Step 6:
Combine the solutions to get the final answer: $w = 0$ or $w = -18$.
Knowledge Notes:
To solve the quadratic equation $w^2 = -18w$, one can use algebraic manipulation to find the values of $w$ that satisfy the equation. Here are the relevant knowledge points:
Moving Terms: Algebraic equations can be balanced by performing the same operation on both sides. In this case, adding $18w$ to both sides maintains the balance and helps to set the equation to a standard form.
Factoring: Factoring is the process of finding numbers or expressions that multiply together to give the original expression. Here, $w$ is a common factor of both terms on the left side of the equation.
Zero Product Property: If a product of factors equals zero, at least one of the factors must be zero. This property allows us to set each factor in the equation $w(w + 18) = 0$ equal to zero and solve for $w$.
Solving Linear Equations: When we have a simple linear equation like $w + 18 = 0$, we can solve for $w$ by isolating the variable on one side of the equation. This is done by performing inverse operations, such as subtracting $18$ from both sides.
Quadratic Solutions: A quadratic equation can have two solutions, and in this case, the solutions are $w = 0$ and $w = -18$. These solutions are the values of $w$ that make the original equation true.
By understanding these concepts, one can solve a variety of algebraic equations, including quadratic equations like the one presented.
