Use a graphing utility to find or approximate​ solution(s) of the given equation.
10 x plus 2 equals 15 x squared plus 3 x
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer.
A.
The​ solution(s) is/are xequals

enter your response here.
​(Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.)
B.
The equation has no solution

To find the solution(s) of the equation \( 10x + 2 = 15x^2 + 3x \) using a graphing utility, follow these steps:

### Step 1: Rewrite the equation
First, rewrite the equation in standard form:

\[
15x^2 + 3x - 10x - 2 = 0
\]

Simplify it:

\[
15x^2 - 7x - 2 = 0
\]

### Step 2: Graph the equation
Graph the function \( f(x) = 15x^2 - 7x - 2 \) using a graphing utility.

### Step 3: Identify the x-intercepts
The solutions to the equation correspond to the x-values where the graph of \( f(x) \) intersects the x-axis (where \( f(x) = 0 \)).

### Approximate Solution(s)
After graphing, we look for the x-intercepts:

- The graph intersects the x-axis at two points, approximately at \( x = 1 \) and \( x = -\frac{2}{3} \).

### Answer
So, the solutions to the equation are:

\[
x = 1 \quad \text{and} \quad x = -\frac{2}{3}
\]

Select the correct choice:

**A.** The solutions are \( x = 1 \) and \( x = -\frac{2}{3} \).

Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How can we factor the quadratic equation \( 15x^2 - 7x - 2 = 0 \)?
2. What are the steps to graph a quadratic function manually?
3. How can you verify the solutions using the quadratic formula?
4. What does the vertex of the quadratic function represent in the graph?
5. How do changes in coefficients affect the graph of a quadratic function?

**Tip:** Always check the discriminant of a quadratic equation to determine the nature of its roots before graphing.

Learn how to use a graphing utility to solve the equation 10x + 2 = 15x^2 + 3x. This problem involves solving a quadratic equation by graphing, identifying x-intercepts, and selecting correct solutions. Suitable for Grades 9-12.

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