Certainly! Let's solve this quadratic equation step-by-step using the information provided:
The given quadratic equation is:
[tex]\[ 3x^2 + 45x + 24 = 0 \][/tex]
We already have one solution to this equation, which is [tex]\(-14.45\)[/tex]. Our goal is to find the other solution.
### Step 1: Use Vieta's Formulas
According to Vieta's formulas for a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], the sum and product of the roots ([tex]\(r1\)[/tex] and [tex]\(r2\)[/tex]) can be given by:
[tex]\[ r1 + r2 = -\frac{b}{a} \][/tex]
[tex]\[ r1 \times r2 = \frac{c}{a} \][/tex]
For our equation [tex]\(3x^2 + 45x + 24 = 0\)[/tex], the coefficients are:
[tex]\[ a = 3 \][/tex]
[tex]\[ b = 45 \][/tex]
[tex]\[ c = 24 \][/tex]
### Step 2: Calculate the Sum of the Roots
Using Vieta's first formula:
[tex]\[ r1 + r2 = -\frac{b}{a} = -\frac{45}{3} = -15 \][/tex]
### Step 3: Identify the Known Root and Use It to Find the Other Root
We know that one of the roots ([tex]\(r1\)[/tex]) is [tex]\(-14.45\)[/tex]. Let's denote the other root by [tex]\(r2\)[/tex].
From the sum of the roots:
[tex]\[ r1 + r2 = -15 \][/tex]
Substituting [tex]\(r1 = -14.45\)[/tex]:
[tex]\[ -14.45 + r2 = -15 \][/tex]
### Step 4: Solve for the Other Root
To find [tex]\(r2\)[/tex], isolate it on one side of the equation:
[tex]\[ r2 = -15 + 14.45 \][/tex]
[tex]\[ r2 = -0.55 \][/tex]
Therefore, the other solution to the quadratic equation [tex]\(3x^2 + 45x + 24 = 0\)[/tex] is [tex]\(-0.55\)[/tex] when rounded to the hundredths place.
So, the correct answer is:
[tex]\[ \boxed{-0.55} \][/tex]
