question 3
x^2 - 7x + 8 = 0

question 4
x^2 + x - 6 = 0

Let's solve the quadratic equations from Question 3 and Question 4 step-by-step.

### Question 3: Solve \(x^2 - 7x + 8 = 0\)

To solve the quadratic equation, we can use the quadratic formula:

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

For the equation \(x^2 - 7x + 8 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -7\)
- \(c = 8\)

Plug these values into the quadratic formula:

\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}
\]

Simplify:

\[
x = \frac{7 \pm \sqrt{49 - 32}}{2}
\]

\[
x = \frac{7 \pm \sqrt{17}}{2}
\]

So, the solutions are:

\[
x_1 = \frac{7 + \sqrt{17}}{2}, \quad x_2 = \frac{7 - \sqrt{17}}{2}
\]

### Question 4: Solve \(x^2 + x - 6 = 0\)

Again, use the quadratic formula. The coefficients for this equation are:
- \(a = 1\)
- \(b = 1\)
- \(c = -6\)

Substitute these values into the formula:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1}
\]

Simplify:

\[
x = \frac{-1 \pm \sqrt{1 + 24}}{2}
\]

\[
x = \frac{-1 \pm \sqrt{25}}{2}
\]

\[
x = \frac{-1 \pm 5}{2}
\]

The two solutions are:

\[
x_1 = \frac{-1 + 5}{2} = 2, \quad x_2 = \frac{-1 - 5}{2} = -3
\]

### Summary of Solutions

- **Question 3**: \(x = \frac{7 + \sqrt{17}}{2}\) and \(x = \frac{7 - \sqrt{17}}{2}\)
- **Question 4**: \(x = 2\) and \(x = -3\)

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you factorize the quadratic equation \(x^2 - 7x + 8 = 0\)?
2. What are the properties of the quadratic formula?
3. How do you complete the square to solve a quadratic equation?
4. What are the roots of the equation \(x^2 + 4x + 4 = 0\)?
5. How can you determine if a quadratic equation has real or complex roots?

**Tip:** The discriminant (\(b^2 - 4ac\)) in the quadratic formula helps determine the nature of the roots: if it's positive, the equation has two distinct real roots; if it's zero, the equation has exactly one real root (a repeated root); if it's negative, the equation has two complex roots.

Learn how to solve quadratic equations x^2 - 7x + 8 = 0 and x^2 + x - 6 = 0 with detailed step-by-step solutions. Understand the quadratic formula and how to apply it to find the roots of these equations. Suitable for high school students.

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