Answer:
Explanation:
To solve these quadratic equations by factoring, you need to find two numbers that multiply to the constant term (the number without x^2) and add up to the coefficient of the linear term (the number with x). Here are the solutions for each of the equations:
1. x² + 5x + 6 = 0
We need two numbers that multiply to 6 and add up to 5. The numbers are 2 and 3.
So, we can factor the equation as (x + 2)(x + 3) = 0.
Now, set each factor equal to zero and solve for x:
x + 2 = 0 => x = -2
x + 3 = 0 => x = -3
So, the solutions are x = -2 and x = -3.
2. x² + 10x + 21 = 0
We need two numbers that multiply to 21 and add up to 10. The numbers are 7 and 3.
So, we can factor the equation as (x + 7)(x + 3) = 0.
Now, set each factor equal to zero and solve for x:
x + 7 = 0 => x = -7
x + 3 = 0 => x = -3
So, the solutions are x = -7 and x = -3.
3. x² + 8x + 15 = 0
We need two numbers that multiply to 15 and add up to 8. The numbers are 5 and 3.
So, we can factor the equation as (x + 5)(x + 3) = 0.
Now, set each factor equal to zero and solve for x:
x + 5 = 0 => x = -5
x + 3 = 0 => x = -3
So, the solutions are x = -5 and x = -3.
4. x² + 9x + 14 = 0
We need two numbers that multiply to 14 and add up to 9. The numbers are 7 and 2.
So, we can factor the equation as (x + 7)(x + 2) = 0.
Now, set each factor equal to zero and solve for x:
x + 7 = 0 => x = -7
x + 2 = 0 => x = -2
So, the solutions are x = -7 and x = -2.
5. x² - 2x - 35 = 0
To factor this equation, we need two numbers that multiply to -35 and add up to -2. The numbers are -7 and 5.
So, we can factor the equation as (x - 7)(x + 5) = 0.
Now, set each factor equal to zero and solve for x:
x - 7 = 0 => x = 7
x + 5 = 0 => x = -5
So, the solutions are x = 7 and x = -5.