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Chapter 11: Problem 39

Write a quadratic equation having the given numbers as solutions. $$ 2.4,-0.4 $$

Short Answer

Expert verified
The quadratic equation is \( x^2 - 2x - 0.96 = 0 \).

Step by step solution

01

Recall the Standard Form of a Quadratic Equation

Recall that a quadratic equation with roots \( r_1 \) and \( r_2 \) can be expressed as: \[ (x - r_1)(x - r_2) = 0 \]
02

Substitute the Given Roots into the Equation

Substitute the given roots \( r_1 = 2.4 \) and \( r_2 = -0.4 \) into the quadratic equation: \[ (x - 2.4)(x + 0.4) = 0 \]
03

Expand the Binomials

Use the distributive property to expand the product of the binomials: \[ (x - 2.4)(x + 0.4) = x^2 + 0.4x - 2.4x - 0.96 \]
04

Simplify the Equation

Combine like terms to obtain the quadratic equation in standard form: \[ x^2 - 2x - 0.96 = 0 \]

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Key Concepts

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

Standard Form of a Quadratic Equation
To start understanding quadratic equations, we need to understand their standard form. The standard form of a quadratic equation is expressed as: \[ ax^2 + bx + c = 0 \]Here, 'a', 'b', and 'c' are constants, and 'x' represents the variable.
This form is useful for identifying the properties of the quadratic function, such as its roots (solutions), vertex, and the direction of the parabola it defines.
When we know the values of 'r1' and 'r2' (the roots of the equation), we can write the equation as:
\[ (x - r_1)(x - r_2) = 0 \].
This will help us understand the relationship between the roots and the coefficients in the standard form of the quadratic equation.
Roots of a Quadratic Equation
The roots of a quadratic equation are the values of 'x' at which the equation equals zero.
For the quadratic equation in standard form, \[ ax^2 + bx + c = 0 \],
the roots can be found using various methods:
  • Factoring
  • Completing the Square
  • Quadratic Formula \[ x = \frac{{-b \, \pm \, \sqrt{{b^2 - 4ac}}}}{2a} \]

In the problem, we are given the roots 2.4 and -0.4. Using these roots, we construct the equation in factored form:
\[ (x - 2.4)(x + 0.4) = 0 \].
Expanding Binomials
Expanding binomials means multiplying them to convert the equation from factored form to standard form.
For our problem, we start with \[ (x - 2.4)(x + 0.4) = 0 \].
Using the distributive property, we multiply each term in the first binomial by each term in the second binomial:
\[ (x - 2.4)(x + 0.4) = x^2 + 0.4x - 2.4x - 0.96 \]
Then, combine like terms (0.4x and -2.4x) to simplify:
\[ x^2 - 2x - 0.96 = 0 \]
Now, the equation is in standard form.
The Distributive Property
The distributive property is a basic algebraic property used to expand expressions.
It states that for any numbers a, b, and c: \[ a(b + c) = ab + ac \]
In our exercise, we applied the distributive property to the binomials: \[ (x - 2.4)(x + 0.4) \]
First, distribute each term in the first binomial to every term in the second binomial:
  • \[ x \, * \, x = x^2 \]
  • \[ x \, * \, 0.4 = 0.4x \]
  • \[ -2.4 \, * \, x = -2.4x \]
  • \[ -2.4 \, * \, 0.4 = -0.96 \]

Next, combine like terms to get the final result: \[ x^2 + 0.4x - 2.4x - 0.96 \rightarrow x^2 - 2x - 0.96 \].
This completes the expansion using the distributive property.

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