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Chapter 0: Problem 59

$$\text { solve each quadratic equation by factoring.}$$ $$5 x^{2}=20 x$$

Short Answer

Expert verified
The solutions of the quadratic equation \(5x^2 = 20x\) are \(x = 0\) and \(x = 4\).

Step by step solution

01

Rearrange the Equation

Rearrange the given equation to a standard form of a quadratic equation which is \(ax^2 + bx + c = 0\). Subtract \(20x\) from both sides of the equation to get \(5x^2 - 20x = 0\).
02

Factoring

Factor the equation by taking out the common factor. Here, \(5x\) is the common factor. Factoring gives \(5x(x - 4) = 0\).
03

Solve the factored equation

Set each factor equal to zero and solve for \(x\). So the solutions are \(x = 0\) and \(x = 4\).

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Key Concepts

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

Factoring Quadratics
Factoring quadratics involves rewriting a quadratic expression as a product of two simpler expressions. When dealing with quadratic equations, it's often easier to solve them if you can break them down into these simpler parts. In the equation from the exercise, after arranging it in standard form, we have \(5x(x - 4) = 0\).

Here's the process for factoring quadratics:
  • Identify common factors: Look at each term in the quadratic expression to see if there's a number or variable that can be divided out. In our exercise, \(5x\) can be factored from both terms.

  • Factor out the common factor: Write the expression in terms of this common factor. This simplifies the equation significantly, making it easier to solve.

  • Finish factoring if necessary: Sometimes you may need to factor further. For quadratics that don't have an obvious common factor, you might have to use techniques like the "ac method" or trial and error to factor them fully.
Factoring helps us simplify quadratics so the zero product property can be used next. It's a crucial skill to master when solving equations efficiently.
Quadratic Equation Standard Form
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This form is very helpful because it allows us to apply various algebraic methods to find the solutions. To get a proper start, converting any given quadratic equation to this standard form is important.

In our original equation, \(5x^2 = 20x\), it's not immediately in standard form as it misses the \(c\) term and is not set equal to zero. Let's see how we do this conversion:
  • Rearrange the terms: Move all terms to one side of the equation to form \(ax^2 + bx = 0\). In our exercise, subtract \(20x\) from both sides.

  • Standardize the layout: Ensure that all like terms (\(x^2\), \(x\), and constants) are correctly placed and combined to form the standard quadratic format.
Converting to standard form is vital as it sets the stage for other techniques like factoring and using the quadratic formula.
Zero Product Property
The zero product property is a fundamental concept used in solving equations, especially quadratics. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. This property is our key to finding solutions once we've successfully factored a quadratic equation.

In the example from the exercise, we reached a factored form \(5x(x - 4) = 0\). This allows us to apply the zero product property:
  • Set each factor to zero: If the product is zero, either \(5x = 0\) or \(x - 4 = 0\) must be true.

  • Solve each equation individually: Solving \(5x = 0\) gives \(x = 0\), and solving \(x - 4 = 0\) gives \(x = 4\). These are the solutions to the quadratic equation.
Utilizing the zero product property lets us break down and approach solutions logically and step-wise. This is why achieving a factored form is desired鈥攊t directly enables the application of this useful property.

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Most popular questions from this chapter

Exercises \(142-144\) will help you prepare for the material covered in the next section. $$\text { Multiply: }\left(2 x^{3} y^{2}\right)\left(5 x^{4} y^{7}\right)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I prefer interval notation over set-builder notation because it takes less space to write solution sets.

What is a linear equation in one variable? Give an example of this type of equation.

Exercises \(142-144\) will help you prepare for the material covered in the next section. Use the distributive property to multiply: $$ 2 x^{4}\left(8 x^{4}+3 x\right) $$

Place the correct symbol, \(>\) or \(<\), in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. \(a, 3^{\frac{1}{2}}\square 3^{1}\) b. \(\sqrt{7}+\sqrt{18} \square \sqrt{7+18}\)
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