Use the quadratic formula to solve the equation.
7equals=−4x squaredx2minus−x
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Part 1
The solution set is
StartSet nothing EndSetenter your response here.
​(Simplify your answer. Type an exact​ answer, using radicals and
ii
as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)

To solve the equation \( 7 = -4x^2 - x \), we first rewrite it in standard quadratic form:

\[
-4x^2 - x - 7 = 0.
\]

### Step 1: Standardize the quadratic equation
Multiply through by \(-1\) to make the coefficient of \(x^2\) positive:
\[
4x^2 + x + 7 = 0.
\]

### Step 2: Apply the quadratic formula
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where \(a = 4\), \(b = 1\), and \(c = 7\).

Substitute these values into the formula:
\[
x = \frac{-1 \pm \sqrt{1^2 - 4(4)(7)}}{2(4)}.
\]

### Step 3: Simplify under the square root
\[
x = \frac{-1 \pm \sqrt{1 - 112}}{8}.
\]
\[
x = \frac{-1 \pm \sqrt{-111}}{8}.
\]

### Step 4: Handle the negative discriminant
Since the discriminant (\(-111\)) is negative, the solutions will be complex. Rewrite the square root:
\[
\sqrt{-111} = i\sqrt{111}.
\]

Thus, the solutions are:
\[
x = \frac{-1 \pm i\sqrt{111}}{8}.
\]

### Final Solution:
The solution set is:
\[
\boxed{\left\{ \frac{-1 + i\sqrt{111}}{8}, \frac{-1 - i\sqrt{111}}{8} \right\}}
\]

Let me know if you'd like further clarification or details!  

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### Related Questions:
1. How does the discriminant help determine the nature of the roots of a quadratic equation?
2. Can a quadratic equation have only one solution? Under what conditions?
3. What is the geometric interpretation of complex roots in a quadratic equation?
4. How do you rewrite a quadratic equation if the leading coefficient is negative?
5. What are the steps to factorize quadratic equations if they can be factored?

**Tip:** Always check the discriminant first to determine whether the roots are real or complex before solving a quadratic equation.

Use the quadratic formula to solve the equation. 7 = -4x^2 - x. The solution set is (Simplify your answer. Type an exact answer, using radicals and i as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

Explore how to solve the quadratic equation 7 = -4x^2 - x using the quadratic formula. Includes detailed steps to identify complex roots through discriminant analysis, with a focus on algebra and complex numbers. Perfect for students in Grades 9-12.

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