The task is to rewrite the quadratic expression $x^2 + 4x - 7$ in the form $(x + a)^2 - b$, where $a$ and $b$ are integers. Here's how to solve it step by step:

Step 1: Start with the quadratic expression

$x^2 + 4x - 7$

Step 2: Complete the square

To complete the square, focus on the terms $x^2 + 4x$:

1. Take half of the coefficient of $x$ (which is $4$) and square it: $\left(\frac{4}{2}\right)^2 = 4$
2. Add and subtract $4$ within the expression: $x^2 + 4x = (x^2 + 4x + 4) - 4 = (x + 2)^2 - 4$

Step 3: Combine with the constant term $-7$

Now substitute back into the full expression: $x^2 + 4x - 7 = (x + 2)^2 - 4 - 7$

Simplify: $(x + 2)^2 - 11$

Final Answer:

$(x + a)^2 - b = (x + 2)^2 - 11$ Here:

• $a = 2$
• $b = 11$

Would you like further clarification or additional examples?

Related Questions:

1. What is the process of completing the square for $x^2 + 6x + 5$?
2. How do you rewrite $x^2 - 8x + 10$ in the form $(x + a)^2 - b$?
3. Why is completing the square helpful in solving quadratic equations?
4. How do you find the vertex of a parabola using the completed square form?
5. Can this method be used for quadratic equations with fractional coefficients?

Tip:

When completing the square, always remember to balance your equation by adding and subtracting the same value!