Solve x^2lessthanorequalto15x-34 algebraically

We are tasked with solving the inequality:

\[
x^2 \leq 15x - 34
\]

### Step 1: Rearrange the inequality
Move all terms to one side to set the inequality to zero:

\[
x^2 - 15x + 34 \leq 0
\]

### Step 2: Solve the corresponding equation
Solve the quadratic equation:

\[
x^2 - 15x + 34 = 0
\]

We will use the quadratic formula:

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

For the equation \(x^2 - 15x + 34 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -15\)
- \(c = 34\)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(1)(34)}}{2(1)}
\]

\[
x = \frac{15 \pm \sqrt{225 - 136}}{2}
\]

\[
x = \frac{15 \pm \sqrt{89}}{2}
\]

Thus, the two solutions are:

\[
x_1 = \frac{15 + \sqrt{89}}{2} \quad \text{and} \quad x_2 = \frac{15 - \sqrt{89}}{2}
\]

These values represent the points where the quadratic expression equals zero.

### Step 3: Analyze the inequality
Now, we need to determine where the quadratic expression \(x^2 - 15x + 34\) is less than or equal to zero.

Since the quadratic is a parabola opening upwards (the coefficient of \(x^2\) is positive), the expression will be less than or equal to zero between the two roots.

### Step 4: Conclusion
The solution to the inequality is the interval between the two roots:

\[
\frac{15 - \sqrt{89}}{2} \leq x \leq \frac{15 + \sqrt{89}}{2}
\]

This is the algebraic solution.

Would you like further details or have any questions?

Here are five related questions to expand your understanding:
1. How do you interpret the graph of a quadratic inequality?
2. What does the discriminant tell you about the number of real solutions?
3. How would the solution change if the inequality were strict (\(<\) instead of \(\leq\))?
4. What are the implications of a quadratic expression always being positive or negative?
5. How would the inequality change if the quadratic expression had complex roots?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) first to know whether you will have real or complex solutions when solving quadratics.

Learn how to solve the quadratic inequality x^2 ≤ 15x - 34 algebraically with a detailed step-by-step solution. This problem involves key algebraic concepts such as quadratic equations and inequalities, and uses the quadratic formula to find the solution. Suitable for students in Grades 9-11.

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