The question provided involves solving the quadratic equation $x^2 - 10x + 2 = 0$ by completing the square and expressing the solutions in the form $a \pm \sqrt{b}$, where $a$ and $b$ are prime numbers.

Step-by-Step Solution:

1. Rewrite the equation: $x^2 - 10x + 2 = 0$

2. Move the constant to the other side: $x^2 - 10x = -2$

3. Complete the square: To complete the square, take half the coefficient of $x$, square it, and add it to both sides. The coefficient of $x$ is $-10$, so: $\left( \frac{-10}{2} \right)^2 = 25$ Add 25 to both sides: $x^2 - 10x + 25 = -2 + 25$ Simplifies to: $(x - 5)^2 = 23$

4. Solve for $x$: Take the square root of both sides: $x - 5 = \pm \sqrt{23}$ Therefore: $x = 5 \pm \sqrt{23}$

5. Interpret the solution: The solutions are $x = 5 \pm \sqrt{23}$, which can be written in the form $a \pm \sqrt{b}$ where $a = 5$ and $b = 23$.

Since $5$ and $23$ are both prime numbers, this satisfies the condition.

Final Answer:

• $a = 5$
• $b = 23$

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

Related Questions:

1. What is the process of completing the square for any general quadratic equation?
2. How can you check if a number is prime, and why is that important in this question?
3. How do you solve quadratic equations using the quadratic formula instead of completing the square?
4. Can the discriminant be used to determine the number and type of solutions for a quadratic equation?
5. What are some real-life applications of solving quadratic equations?

Tip:

When completing the square, always ensure you balance the equation by adding the same value to both sides.