Let's solve the problem step-by-step.
1. Identify the coefficients from the quadratic equation:
The given quadratic equation is:
[tex]\[
y = 3x^2 + 5x + 1
\][/tex]
From this equation, we can identify the coefficients:
[tex]\[
a = 3, \quad b = 5, \quad c = 1
\][/tex]
2. Recall the quadratic formula:
The quadratic formula to find the roots of a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Substitute the coefficients into the quadratic formula:
In our case, substituting [tex]\( a = 3 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = 1 \)[/tex]:
[tex]\[
x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}
\][/tex]
4. Determine the number in the green box:
The expression under the square root [tex]\(b^2 - 4ac\)[/tex] simplifies to [tex]\(5^2 - 4 \cdot 3 \cdot 1\)[/tex], which we've identified as the value 13. However, the question asks for the value to put into the green box, which references the first part of the discriminant [tex]\(b\)[/tex].
Therefore, the correct substitution for determining the number to place in the green box is:
[tex]\[
x = \frac{-5 \pm \sqrt{5^2 - 4(3)(1)}}{2(3)}
\][/tex]
So, the number that belongs in the green box is:
[tex]\[
5
\][/tex]
Thus, the correct answer is:
[tex]\[
5
\][/tex]
