https://file.aiquickdraw.com/mathbot/file/17263424233887a9wvaq3.jpg

We are given the quadratic equation:

\[
x^2 - 3x + 1 = 0
\]

with \( m \) and \( n \) being the roots. We are tasked to find the numerical value of:

\[
2m^2 + 4n^2 - 6n + 2006
\]

### Step 1: Use Vieta's Formulas
For a quadratic equation \( ax^2 + bx + c = 0 \), Vieta's formulas tell us that:
- The sum of the roots is \( m + n = -\frac{b}{a} \), and
- The product of the roots is \( mn = \frac{c}{a} \).

For the given quadratic equation \( x^2 - 3x + 1 = 0 \):
- \( m + n = 3 \),
- \( mn = 1 \).

### Step 2: Find \( m^2 + n^2 \)
We need to compute \( m^2 + n^2 \) as it is part of the expression we need to evaluate.

We know:

\[
m^2 + n^2 = (m + n)^2 - 2mn
\]

Substitute the values for \( m + n \) and \( mn \):

\[
m^2 + n^2 = (3)^2 - 2(1) = 9 - 2 = 7
\]

### Step 3: Simplify the Expression
Now, we substitute \( m^2 + n^2 = 7 \) into the given expression \( 2m^2 + 4n^2 - 6n + 2006 \).

We know \( m^2 + n^2 = 7 \), so we can split this expression further:

\[
2m^2 + 4n^2 = 2(m^2 + 2n^2) = 2(7 + n^2)
\]

Thus the expression simplifies into  `complete verification

If m and n are two roots of x^2 - 3x + 1 = 0, then what is the numerical value of 2m^2 + 4n^2 - 6n + 2006?

Explore how to use Vieta's formulas to solve for the value of 2m^2 + 4n^2 - 6n + 2006, given that m and n are roots of the quadratic equation x^2 - 3x + 1 = 0. Includes a step-by-step breakdown and detailed algebraic simplifications. Suitable for Grades 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation