Let f(×) = 3x +1 and g(×) = 2-x. Determine the values for a such that f(a^2) = g(2a)

To find the values of \(a\) such that \(f(a^2) = g(2a)\), we need to evaluate the given functions \(f(x) = 3x + 1\) and \(g(x) = 2 - x\) for the specified inputs and then solve the resulting equation.

### Step 1: Evaluate \(f(a^2)\) and \(g(2a)\)

Given:
\[
f(x) = 3x + 1
\]
\[
g(x) = 2 - x
\]

Evaluate \(f(a^2)\):
\[
f(a^2) = 3(a^2) + 1 = 3a^2 + 1
\]

Evaluate \(g(2a)\):
\[
g(2a) = 2 - (2a) = 2 - 2a
\]

### Step 2: Set up the equation

Now, set \(f(a^2)\) equal to \(g(2a)\):
\[
3a^2 + 1 = 2 - 2a
\]

### Step 3: Rearrange the equation

Rearranging the terms to one side, we get a quadratic equation:
\[
3a^2 + 2a + 1 - 2 = 0
\]
\[
3a^2 + 2a - 1 = 0
\]

### Step 4: Solve the quadratic equation

The quadratic equation \(3a^2 + 2a - 1 = 0\) can be solved using the quadratic formula:
\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \(a = 3\), \(b = 2\), and \(c = -1\).

Substitute these values into the formula:
\[
a = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3}
\]

Simplify inside the square root:
\[
a = \frac{-2 \pm \sqrt{4 + 12}}{6}
\]
\[
a = \frac{-2 \pm \sqrt{16}}{6}
\]
\[
a = \frac{-2 \pm 4}{6}
\]

### Step 5: Find the roots

Now, solve for the two possible values of \(a\):

1. \(a = \frac{-2 + 4}{6} = \frac{2}{6} = \frac{1}{3}\)

2. \(a = \frac{-2 - 4}{6} = \frac{-6}{6} = -1\)

### Conclusion

The values of \(a\) that satisfy the equation \(f(a^2) = g(2a)\) are:

\[
\boxed{a = \frac{1}{3} \text{ and } a = -1.}
\]

Would you like further details on any part of the solution or have any questions?

### Related Questions:

1. How would the solution change if the functions were different?
2. What are the properties of quadratic functions that we used to solve this problem?
3. Can we graph the functions \(f(x)\) and \(g(x)\) to visually find the solution?
4. What are the methods to solve a quadratic equation other than the quadratic formula?
5. How do the values of \(a\) affect the outputs of \(f(x)\) and \(g(x)\)?

### Tip: 

Always check your solutions by substituting them back into the original equations to ensure they satisfy all conditions.

Let f(x) = 3x +1 and g(x) = 2 - x. Determine the values for a such that f(a^2) = g(2a)

Discover how to solve the equation f(a^2) = g(2a) where f(x) = 3x + 1 and g(x) = 2 - x, leading to a quadratic equation. Learn key algebraic concepts and how to apply the quadratic formula to find the values of a. This solution is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation