https://file.aiquickdraw.com/mathbot/file/17268784499014cfknzlk.jpg

The inequality given in the image is:

\[
2x^2 + 4 \leq 8x
\]

To solve this inequality, we will follow these steps:

### Step 1: Move all terms to one side
We want to bring all terms to one side of the inequality:

\[
2x^2 + 4 - 8x \leq 0
\]

### Step 2: Simplify the expression
We now have:

\[
2x^2 - 8x + 4 \leq 0
\]

### Step 3: Factor the quadratic expression
Factor out the common factor of 2:

\[
2(x^2 - 4x + 2) \leq 0
\]

Now we need to factor the quadratic \(x^2 - 4x + 2\), but since it doesn’t factor nicely, we will use the quadratic formula to find the roots:

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

Simplifying the discriminant:

\[
x = \frac{4 \pm \sqrt{16 - 8}}{2} = \frac{4 \pm \sqrt{8}}{2} = \frac{4 \pm 2\sqrt{2}}{2}
\]

Thus, the solutions are:

\[
x = 2 \pm \sqrt{2}
\]

So the two roots are:

\[
x_1 = 2 + \sqrt{2}, \quad x_2 = 2 - \sqrt{2}
\]

### Step 4: Analyze the quadratic inequality
Now, we need to analyze the sign of the expression \(2(x^2 - 4x + 2)\) between the roots \(x_1 = 2 + \sqrt{2}\) and \(x_2 = 2 - \sqrt{2}\). Since the parabola opens upwards (the coefficient of \(x^2\) is positive), the quadratic will be less than or equal to 0 between these two roots.

### Step 5: Write the solution in interval notation
Thus, the solution to the inequality \(2x^2 + 4 \leq 8x\) is the interval between \(x_1\) and \(x_2\):

\[
[2 - \sqrt{2}, 2 + \sqrt{2}]
\]

This is the solution in interval notation.

---

Would you like any further details or have any questions on any of these steps?

Here are five related questions you might find helpful:

1. How do you solve a quadratic inequality graphically?
2. What is the significance of the discriminant in the quadratic formula?
3. Can all quadratic inequalities be factored without using the quadratic formula?
4. How do you check if a quadratic inequality includes the endpoints?
5. What are the differences between solving an equation and solving an inequality?

**Tip:** When solving inequalities, remember that multiplying or dividing by a negative number flips the inequality sign.

Solve the inequality and write the solution in interval notation. Leave answers in exact, simplified form. 2x^2 + 4 ≤ 8x

Explore the step-by-step process to solve the quadratic inequality 2x^2 + 4 ≤ 8x with interval notation for solutions. This guide covers factoring, using the quadratic formula, and testing signs of expressions. Ideal for high school algebra students.

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