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Chapter 5: Problem 21

Solve each equation. $$ 6 x^{2}-36 x=0 $$

Short Answer

Expert verified
x = 0, x = 6

Step by step solution

01

Factor out the common term

The given equation is n6 x^{2} - 36 x = 0.Factor out the common term, which is 6x: 6x(x - 6) = 0.
02

Use the zero product property

Set each factor equal to zero: 6x = 0 and x - 6 = 0.
03

Solve each equation

Solve for x in both equations. For 6x = 0, divide both sides by 6: x = 0. For x - 6 = 0, add 6 to both sides: x = 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring
Factoring is the process of breaking down an equation into simpler components (factors) that, when multiplied together, give the original equation. In the given problem, the original quadratic equation is: 6x^2 - 36x = 0To factor this, first look for a common factor shared by all terms. In this case, both terms (6x^2 and -36x) have a common factor of 6x.When we factor out 6x, the equation becomes:6x(x - 6) = 0Now, the quadratic equation is factored into two simpler elements: 6x and (x - 6). These elements are easier to work with when we move to the next steps.
Zero Product Property
The Zero Product Property is a fundamental rule in algebra that says if a product of two factors is zero, then at least one of the factors must be zero. We use this property to solve factored polynomial equations. So, if we have an equation in the form of: A * B = 0Then either A = 0 or B = 0 (or both).In our example, after factoring, we have: 6x(x - 6) = 0To find the solutions, we set each factor to zero:
  • 6x = 0
  • x - 6 = 0
By solving these simpler equations, we can find the values of x that satisfy the original equation.
Solving Polynomial Equations
Solving polynomial equations involves finding the variable values that satisfy the equation. After factoring and applying the zero product property, we solve for x.In our example, we get two simple equations to solve:
  • 6x = 0
  • x - 6 = 0
Let's solve each one step by step.For the equation 6x = 0, we divide both sides by 6:x = 0For the equation x - 6 = 0, we add 6 to both sides:x = 6Thus, the solutions to the original equation are x = 0 and x = 6. These are the values for the variable x that make the original quadratic equation true.

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