To factorize the quadratic expression $8 - x - 9x^2$, we will follow a systematic approach.

First, rewrite the expression in standard form: $-9x^2 - x + 8$

Next, we look for two numbers that multiply to $a \cdot c = (-9) \cdot 8 = -72$ and add up to $b = -1$.

The numbers that satisfy these conditions are -9 and 8 because: $(-9) \cdot 8 = -72$ $(-9) + 8 = -1$

Using these numbers, we can rewrite the middle term (-x) as a combination of these factors: $-9x^2 - 9x + 8x + 8$

Now, group the terms: $(-9x^2 - 9x) + (8x + 8)$

Factor out the common factors from each group: $-9x(x + 1) + 8(x + 1)$

Notice that $(x + 1)$ is a common factor: $(-9x + 8)(x + 1)$

So, the factorized form of the quadratic expression $8 - x - 9x^2$ is: $(8 - 9x)(x + 1)$

This gives us the final factorized expression: $8 - x - 9x^2 = (8 - 9x)(x + 1)$

Would you like more details or have any questions?

Suggested Questions:

1. How do you determine which numbers to use for factoring a quadratic expression?
2. What is the grouping method in factoring quadratic expressions?
3. Can you explain how to check if the factorization of a quadratic expression is correct?
4. How do you solve a quadratic equation by factoring?
5. What are some techniques for factoring quadratic expressions that are not easily factorable?

Tip:

When factoring quadratics, always check your factorization by expanding the factors back out to ensure they produce the original expression. This verifies your work and helps catch any mistakes.