To rewrite the quadratic equation $x^2 - 8x + 5 = 0$ in the form $(x + a)^2 + b = 0$, we will complete the square. Here's the step-by-step process:

1. Start with the equation:

$x^2 - 8x + 5 = 0$

2. Move the constant term (5) to the other side:

$x^2 - 8x = -5$

3. Complete the square for the expression $x^2 - 8x$:

• Take half of the coefficient of $x$, which is $-8$, so half of $-8$ is $-4$.
• Square $-4$ to get $16$.

Add 16 to both sides of the equation to keep the equality balanced: $x^2 - 8x + 16 = -5 + 16$

This simplifies to: $(x - 4)^2 = 11$

4. Final result:

So, the quadratic equation $x^2 - 8x + 5 = 0$ can be rewritten as: $(x - 4)^2 - 11 = 0$

This is in the desired form $(x + a)^2 + b = 0$, where $a = -4$ and $b = -11$.

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Would you like further clarification or have any specific questions? Here are some related questions to explore:

1. What does "completing the square" mean in general?
2. How can completing the square be used to find the roots of a quadratic equation?
3. What happens if the coefficient of $x^2$ is not 1 in a quadratic equation?
4. How do you solve quadratic equations by using the quadratic formula?
5. How does the method of completing the square relate to the vertex form of a quadratic function?

Tip: Completing the square is especially useful when you need to convert a quadratic equation into vertex form, which gives you the vertex of the parabola directly.