Solve the inequality. (Enter your answer using interval notation.)

−2*x*2 + 2*x* ≤ −25

Let's solve the inequality step by step:

The given inequality is:

\[
-2x^2 + 2x \leq -25
\]

### Step 1: Move all terms to one side.
Add 25 to both sides to bring everything to one side:

\[
-2x^2 + 2x + 25 \leq 0
\]

### Step 2: Divide through by -2.
To simplify the expression, divide the entire inequality by -2. Remember, dividing by a negative number reverses the inequality sign:

\[
x^2 - x - \frac{25}{2} \geq 0
\]

### Step 3: Solve the quadratic equation.

At this point, solving the quadratic equation would involve the discriminant:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-\frac{25}{2})}}{2(1)}
\]

This expression simplifies to find the roots. However, it looks a bit complicated and likely involves a non-factorable quadratic.

Let me check this more carefullyThe solution to the inequality \( x^2 - x - \frac{25}{2} \geq 0 \) is:

\[
x \in (-\infty, -3.07] \cup [4.07, \infty)
\]

This means that the inequality holds for \( x \) values less than or equal to approximately \(-3.07\), and for \( x \) values greater than or equal to approximately \(4.07\).

### Do you want more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the solution change if the inequality were strict \( > \) instead of \( \geq \)?
2. What are the exact roots of the quadratic equation?
3. How do we graph the inequality on a number line?
4. What is the role of the discriminant in solving quadratic inequalities?
5. Can we apply this approach to solving cubic inequalities?

**Tip:** Always remember that dividing an inequality by a negative number reverses the inequality sign!

Solve the inequality. (Enter your answer using interval notation.) −2*x^2 + 2*x ≤ −25

Learn how to solve the quadratic inequality −2x^2 + 2x ≤ −25 step by step. This tutorial covers key algebraic concepts like quadratic equations, discriminants, and interval notation, and is suitable for students in Grades 9-11.

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