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Part 1
Use a graphing utility to find or approximate​ solution(s) of the given equation.
16 x plus 4 equals 20 x squared plus 5 x
Question content area bottom
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.

We are tasked with solving the equation:

\[
16x + 4 = 20x^2 + 5x
\]

### Step 1: Rearrange the equation into standard form
First, move all terms to one side of the equation to set it equal to 0:

\[
16x + 4 - 20x^2 - 5x = 0
\]

Simplifying this:

\[
-20x^2 + 11x + 4 = 0
\]

This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \) where \( a = -20 \), \( b = 11 \), and \( c = 4 \).

### Step 2: Solve using the quadratic formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values \( a = -20 \), \( b = 11 \), and \( c = 4 \) into the quadratic formula:

\[
x = \frac{-11 \pm \sqrt{11^2 - 4(-20)(4)}}{2(-20)}
\]

\[
x = \frac{-11 \pm \sqrt{121 + 320}}{-40}
\]

\[
x = \frac{-11 \pm \sqrt{441}}{-40}
\]

\[
x = \frac{-11 \pm 21}{-40}
\]

### Step 3: Find the solutions
Now, we calculate both possible solutions:

1. Using the positive square root:
   \[
   x_1 = \frac{-11 + 21}{-40} = \frac{10}{-40} = -\frac{1}{4}
   \]

2. Using the negative square root:
   \[
   x_2 = \frac{-11 - 21}{-40} = \frac{-32}{-40} = \frac{4}{5}
   \]

### Step 4: Conclusion
The solutions are:

\[
x = -\frac{1}{4}, \frac{4}{5}
\]

Thus, the correct answer is:

**A.** The solution(s) is/are \( x = -\frac{1}{4}, \frac{4}{5} \).

Would you like more details, or do you have any other questions?

---

Here are 5 related questions you might consider:
1. How can we verify solutions of a quadratic equation?
2. What is the significance of the discriminant in the quadratic formula?
3. How does the graph of a quadratic equation look for \( a < 0 \)?
4. Can all quadratic equations be solved using factoring instead of the quadratic formula?
5. How do you graph quadratic functions using their vertex and intercepts?

**Tip**: Always check your solutions by substituting them back into the original equation to ensure they satisfy it.

Use a graphing utility to find or approximate​ solution(s) of the given equation. 16x + 4 = 20x^2 + 5x.

Discover how to solve the quadratic equation 16x + 4 = 20x^2 + 5x using the quadratic formula. This detailed guide explores key algebraic concepts and formulas, ideal for students in Grades 9-11.

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