Let's solve the given quadratic equation step by step:
1. Move all terms to one side of the equation to set it to zero:
[tex]\[ x^2 + 17x + 72 = 12 \][/tex]
Subtract 12 from both sides:
[tex]\[ x^2 + 17x + 72 - 12 = 0 \][/tex]
Simplifies to:
[tex]\[ x^2 + 17x + 60 = 0 \][/tex]
2. Factor the quadratic equation [tex]\( x^2 + 17x + 60 = 0 \)[/tex]:
To factor this, we need to find two numbers that multiply to 60 and add up to 17.
The possible pairs of factors of 60 are:
- [tex]\( 1 \times 60 \)[/tex]
- [tex]\( 2 \times 30 \)[/tex]
- [tex]\( 3 \times 20 \)[/tex]
- [tex]\( 4 \times 15 \)[/tex]
- [tex]\( 5 \times 12 \)[/tex]
- [tex]\( 6 \times 10 \)[/tex]
Among these pairs, the pair that adds up to 17 is [tex]\(5\)[/tex] and [tex]\(12\)[/tex].
So, we factor the quadratic as:
[tex]\[ (x + 5)(x + 12) = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
If [tex]\( (x + 5)(x + 12) = 0 \)[/tex], then we set each factor equal to zero:
[tex]\[ x + 5 = 0 \][/tex]
[tex]\[ x = -5 \][/tex]
[tex]\[ x + 12 = 0 \][/tex]
[tex]\[ x = -12 \][/tex]
So, the solutions to the equation are [tex]\( x = -5 \)[/tex] and [tex]\( x = -12 \)[/tex].
Now, we know the equation was solved correctly by one of the students, and the solutions are correctly identified.
Please select the correct answer from each drop-down menu:
The equation was solved correctly by [tex]\( \boxed{1} \)[/tex]. The solutions of the equation are [tex]\( \boxed{-5} \)[/tex] and [tex]\( \boxed{-12} \)[/tex].
