Question

the graph of ( y = x^2 + 11x + 24 ) is equivalent to the graph of which equation?
( y = (x + 8)(x + 3) )
( y = (x + 4)(x + 6) )
( y = (x + 9)(x + 2) )
( y = (x + 7)(x + 4) )

Explanation:

Step1: Factor the quadratic equation

We need to factor \( y = x^2 + 11x + 24 \). We look for two numbers that multiply to \( 24 \) and add up to \( 11 \).
The numbers \( 3 \) and \( 8 \) multiply to \( 24 \) (\( 3\times8 = 24 \)) and add up to \( 11 \) (\( 3 + 8 = 11 \)).
So, we can factor the quadratic as \( y=(x + 3)(x + 8) \) (or \( y=(x + 8)(x + 3) \) since multiplication is commutative).

Step2: Check the options

Now we check the given options:

• Option 1: \( y=(x + 8)(x + 3) \) which matches our factored form.
• Option 2: \( (x + 4)(x + 6)=x^2+10x + 24 \), which is not equal to \( x^2 + 11x + 24 \).
• Option 3: \( (x + 9)(x + 2)=x^2+11x + 18 \), which is not equal to \( x^2 + 11x + 24 \).
• Option 4: \( (x + 7)(x + 4)=x^2+11x + 28 \), which is not equal to \( x^2 + 11x + 24 \).

Answer:

\( y=(x + 8)(x + 3) \) (the first option)