In the image, we have the following equation:

$y = x^2 - 10x + 21$

Chelita attempts to find the $x$-intercepts by solving $y = 0$. She proceeds as follows:

1. Step 1: Set $y = 0$, leading to $x^2 - 10x + 21 = 0$.
2. Step 2: Factor the quadratic expression $x^2 - 10x + 21 = (x + 7)(x + 3)$.
3. Step 3: Set each factor to zero: $x + 7 = 0 \quad \text{or} \quad x + 3 = 0$ Therefore, $x = -7$ or $x = -3$.

However, Chelita's factoring is incorrect. Let's find the correct factors.

Correct Factorization

We start with the quadratic equation:

$x^2 - 10x + 21 = 0$

We need two numbers that multiply to $21$ and add up to $-10$. The correct pair of numbers is $-3$ and $-7$. Thus, the correct factorization is:

$(x - 7)(x - 3) = 0$

Now, set each factor equal to zero:

$x - 7 = 0 \quad \text{or} \quad x - 3 = 0$

Therefore, $x = 7$ or $x = 3$.

Chelita's Error

Chelita mistakenly factored the quadratic as $(x + 7)(x + 3)$, leading to incorrect solutions. The correct solutions are $x = 7$ and $x = 3$, not $x = -7$ and $x = -3$.

Would you like further details or explanations? Here are some related questions:

1. How do you factor a quadratic equation?
2. What are the common mistakes in factoring quadratics?
3. How can we verify the solutions of a quadratic equation?
4. What are other methods to solve quadratic equations besides factoring?
5. What are the $x$- and $y$-intercepts of a quadratic function?

Tip: Always check the product and sum when factoring quadratics to avoid mistakes.