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Chapter 16: Problem 5

Write the quadratic equation in standard form. $$x+5=x(x-3)$$

Short Answer

Expert verified
The standard form of the given quadratic equation is \( x^2 - 4x - 5 = 0 \).

Step by step solution

01

Expand the right side of the equation

Expand the right side of the equation by distributing \( x \) to each term inside the brackets. This gives the equation \( x + 5 = x^2 - 3x \).
02

Rearrange to standard form

Rearrange the terms to get the standard form of the quadratic equation. Subtract \( x \) and 5 from both sides to move all terms to the left side of the equation. This gives \( 0 = x^2 - 4x - 5 \).
03

Revert the equation

In standard form, the equation should have the terms on the left side and \( 0 \) on the right. So, we finally get \( x^2 - 4x - 5 = 0 \).

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Key Concepts

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

Standard Form
When working with quadratic equations, one of the most common tasks is to express the equation in its standard form. The standard form of a quadratic equation is:
  • \( ax^2 + bx + c = 0 \)
where:
  • \( a \), \( b \), and \( c \) are constants,
  • \( x \) is the variable.
The goal of converting to standard form is to arrange all terms with the variable \( x \) on one side and zero on the other.
In this manner, it allows for easier application of solution methods such as factoring, completing the square, or using the quadratic formula. Remember, the power of \( x \) has to be in decreasing order from left to right.
Expanding Expressions
Expanding expressions involves removing parentheses (or brackets) and ensuring that each term is multiplied by any term outside the parentheses. For instance, when you have an expression like \( x(x-3) \), you use the distributive property:
  • Multiply \( x \) by each term inside the parentheses.
This means:
  • \( x \times x = x^2 \)
  • \( x \times (-3) = -3x \)
Combining these, we write \( x(x-3) = x^2 - 3x \).
Expanding is essential as it allows you to see all terms clearly and helps in proceeding steps such as rearranging or solving the equation.
Being careful during expansion is crucial; it's easy to miss a negative sign or overlook a multiplication, which can lead to mistakes in following steps.
Rearranging Equations
Rearranging equations is crucial for writing them in the desired format, such as the standard form.
Once an expression is expanded, ensure all terms are properly consolidated on one side of the equation to maintain balance. This often involves shifting terms from one side to the other. Here's how you go about it:
  • Ensure all terms are accounted for, without leaving any out.
  • Move terms with \( x \) and constants to one side using inverse operations such as addition or subtraction.
  • Confirm that the equation equals zero on one side by applying arithmetic operations.
For the equation \( x + 5 = x^2 - 3x \), rearrange so all terms are on one side:
  • Subtract \( x \) and \( 5 \) from both sides, resulting in \( 0 = x^2 - 4x - 5 \).
  • Or flip it to show \( x^2 - 4x - 5 = 0\).
Ultimately, rearranging aids in setting up the problem for solution techniques and keeps the equation organized.

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