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Chapter 7: Problem 36

Solve the given quadratic equations by factoring. $$V\left(V^{2}-4\right)=V^{2}(V-1)$$

Short Answer

Expert verified
The solutions are \( V = 0 \) and \( V = 4 \).

Step by step solution

01

Expand and Set Equation to Zero

Start by expanding both sides of the equation. Expand the left hand side to get \( V^3 - 4V \). The right hand side is \( V^3 - V^2 \). Therefore, the equation becomes \( V^3 - 4V = V^3 - V^2 \). Subtract \( V^3 \) from both sides to get \( -4V = -V^2 \). Rearrange this to get \( V^2 - 4V = 0 \) by moving all terms to one side.
02

Factor the Quadratic Expression

Factor the quadratic expression \( V^2 - 4V \). We can factor out \( V \) from the expression to get \( V(V - 4) = 0 \). This means the factored form of the equation is \( V(V - 4) = 0 \).
03

Solve for the Variable

Set each factor from the previous step equal to zero and solve for \( V \). First, \( V = 0 \). Second, \( V - 4 = 0 \) which gives \( V = 4 \). Thus, the solutions are \( V = 0 \) and \( V = 4 \).

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Key Concepts

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

Factoring
Factoring is a critical skill when solving quadratic equations, and it involves breaking down an algebraic expression into simpler multiplicative components. In our exercise, the given expression is a polynomial equation that needs to be factored to find solutions. For instance, to factor \( V^2 - 4V \), we look for common factors and simpler expressions that multiply to give the original quadratic.Key Steps to Factor:
  • Identify common factors: For the expression \( V^2 - 4V \), notice that \( V \) is common.
  • Use the distributive property: Factor out the common term to simplify, giving \( V(V - 4) \).
Factoring transforms a complex expression into a product of simpler terms, making it easier to solve for the variable. The goal is always to express the polynomial as a product set to zero, enabling the application of the Zero Product Property.
Solving Quadratic Equations
Solving quadratic equations involves finding the values of the variable that make the equation true. Once an equation is in its factored form, such as \( V(V - 4) = 0 \), we can solve for the variable using the Zero Product Property. This property states that if a product of factors equals zero, at least one of the factors must equal zero.Applying the Zero Product Property:
  • Set each factor equal to zero: \( V = 0 \) or \( V - 4 = 0 \).
  • Solve each equation: The solutions obtained are \( V = 0 \) and \( V = 4 \).
By factoring and applying this property, we can efficiently find solutions to quadratic equations, confirming they satisfy the original polynomial equation once inserted back.
Polynomial Equations
Polynomial equations, such as the one given, consist of variables, coefficients, and exponents combined in an expression set equal to zero. A quadratic equation is a specific type of polynomial equation with a degree of 2, reflecting the highest exponent of the variable is two.Important Characteristics of Polynomial Equations:
  • Contain terms of varying degrees: In our example, \( V^2 \) and \(-4V\) are combined.
  • Can often be simplified and factored: This simplifies solving for unknowns.
  • May require arranging terms: Move all terms to one side before factoring.
Understanding these polynomial structures helps in identifying the appropriate method for solving. By factoring, we simplify their complexity, making the process of finding solutions straightforward and effective.

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Most popular questions from this chapter

Use a calculator to graph all three parabolas on the same coordinate system. Describe (a) the shifts (see page 105 ) of \(y=x^{2}\) that occur and \((b)\) how each parabola opens. (b) \(y=(x-2)^{2}+3 \quad\) (a) \(y=x^{2}\) (c) \(y=(x+2)^{2}-3\)

Without solving the given equations, determine the character of the roots. $$3 x^{2}=14-19 x$$

Solve the given problems. All numbers are accurate to at least two significant digits. A computer monitor has a viewing screen that is \(33.8 \mathrm{cm}\) wide and \(27.3 \mathrm{cm}\) high, with a uniform edge around it. If the edge covers \(20.0 \%\) of the monitor front, what is the width of the edge?

Use a calculator to solve the given equations. Round solutions to the nearest hundredth. If there are no real roots, state this. $$2 t^{2}=7 t+4$$

Use a calculator to solve the given equations. Round solutions to the nearest hundredth. If there are no real roots, state this. $$5-x^{2}=0$$
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