For how many values ​​of a, x=-1 is the solution of the equation? the equation is √x^2+ax+17=ax-2





To solve the given equation \( \sqrt{x^2 + ax + 17} = ax - 2 \) and find the number of values of \( a \) for which \( x = -1 \) is a solution, we will follow these steps:

### Step 1: Substitute \( x = -1 \) into the equation

Substitute \( x = -1 \) into the equation:
\[
\sqrt{(-1)^2 + a(-1) + 17} = a(-1) - 2
\]
This simplifies to:
\[
\sqrt{1 - a + 17} = -a - 2
\]
Further simplifying inside the square root:
\[
\sqrt{18 - a} = -a - 2
\]

### Step 2: Square Both Sides

Square both sides to eliminate the square root:
\[
(\sqrt{18 - a})^2 = (-a - 2)^2
\]
\[
18 - a = (a + 2)^2
\]

### Step 3: Expand and Rearrange the Equation

Expand the right-hand side:
\[
18 - a = a^2 + 4a + 4
\]
Rearrange the equation to bring all terms to one side:
\[
a^2 + 4a + a + 4 - 18 = 0
\]
\[
a^2 + 5a - 14 = 0
\]

### Step 4: Solve the Quadratic Equation

Now, we solve the quadratic equation \( a^2 + 5a - 14 = 0 \) using the quadratic formula:
\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 5 \), and \( c = -14 \). Substituting these values:
\[
a = \frac{-5 \pm \sqrt{5^2 - 4(1)(-14)}}{2(1)}
\]
\[
a = \frac{-5 \pm \sqrt{25 + 56}}{2}
\]
\[
a = \frac{-5 \pm \sqrt{81}}{2}
\]
\[
a = \frac{-5 \pm 9}{2}
\]

This gives two solutions:
\[
a = \frac{-5 + 9}{2} = 2 \quad \text{and} \quad a = \frac{-5 - 9}{2} = -7
\]

### Conclusion

There are \( \boxed{2} \) values of \( a \) for which \( x = -1 \) is a solution to the equation. These values are \( a = 2 \) and \( a = -7 \).

Would you like further details, or do you have any questions?

Here are five related questions to explore further:

1. What are the roots of the quadratic equation \( a^2 + 5a - 14 = 0 \)?
2. How would the solutions change if the constant in the square root were different?
3. What if the equation were \( \sqrt{x^2 + ax + c} = ax - b \)? How would \( a \) and \( b \) interact?
4. How can you determine if a given value of \( x \) is a solution without solving the quadratic?
5. What other values of \( x \) might solve the equation for different values of \( a \)?

**Tip:** When solving equations involving square roots, always remember to check for extraneous solutions, as squaring both sides can introduce solutions that do not satisfy the original equation.

Learn how to solve the equation √x^2 + ax + 17 = ax - 2 and find the values of a for which x = -1 is a solution. This problem involves quadratic equations and square roots, with step-by-step solutions and examples. Suitable for high school students.

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