To solve quadratic equations in the form of x² + bx + c = 0 by factorizing, follow these steps:
a) For the equation p² - p - 42 = 0:
1. First, multiply the coefficient of the x² term (which is 1) by the constant term (which is -42). In this case, 1 * -42 = -42.
2. Find two numbers that multiply to -42 and add up to the coefficient of the x term (which is -1). In this case, the numbers are -7 and 6, because -7 * 6 = -42 and -7 + 6 = -1.
3. Rewrite the middle term -p as the sum of these two numbers: -7p + 6p.
4. Now, rewrite the equation with the grouped terms: p² - 7p + 6p - 42 = 0.
5. Factor by grouping: p(p - 7) + 6(p - 7) = 0.
6. Factor out the common factor (p - 7): (p - 7)(p + 6) = 0.
7. Set each factor equal to zero and solve for p: p - 7 = 0 or p + 6 = 0.
8. Solve for p: p = 7 or p = -6.
b) For the equation r² + 5r - 6 = 0:
1. Multiply the coefficient of the x² term (which is 1) by the constant term (which is -6). In this case, 1 * -6 = -6.
2. Find two numbers that multiply to -6 and add up to the coefficient of the x term (which is 5). In this case, the numbers are 6 and -1, because 6 * -1 = -6 and 6 + (-1) = 5.
3. Rewrite the middle term 5r as the sum of these two numbers: 6r - r.
4. Now, rewrite the equation with the grouped terms: r² + 6r - r - 6 = 0.
5. Factor by grouping: r(r + 6) - 1(r + 6) = 0.
6. Factor out the common factor (r + 6): (r + 6)(r - 1) = 0.
7. Set each factor equal to zero and solve for r: r + 6 = 0 or r - 1 = 0.
8. Solve for r: r = -6 or r = 1.
In summary:
- For the equation p² - p - 42 = 0, the solutions are p = 7 and p = -6.
- For the equation r² + 5r - 6 = 0, the solutions are r = -6 and r = 1.