Sure, let's solve the quadratic equation [tex]\( a^2 + 42a + 432 = 0 \)[/tex] step by step.
### Step 1: Write Down the Standard Form
A quadratic equation is generally written in the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
For our equation, [tex]\( a = 1 \)[/tex], [tex]\( b = 42 \)[/tex], and [tex]\( c = 432 \)[/tex].
### Step 2: Calculate the Discriminant
The discriminant Δ of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the coefficients [tex]\( a = 1 \)[/tex], [tex]\( b = 42 \)[/tex], and [tex]\( c = 432 \)[/tex]:
[tex]\[ \Delta = 42^2 - 4 \cdot 1 \cdot 432 \][/tex]
[tex]\[ \Delta = 1764 - 1728 \][/tex]
[tex]\[ \Delta = 36 \][/tex]
### Step 3: Determine the Nature of the Roots
The discriminant tells us the nature of the roots:
- If [tex]\( \Delta > 0 \)[/tex], there are two distinct real roots.
- If [tex]\( \Delta = 0 \)[/tex], there is exactly one real root.
- If [tex]\( \Delta < 0 \)[/tex], there are no real roots (the roots are complex).
In this case, [tex]\( \Delta = 36 \)[/tex] which is greater than 0. Hence, we have two distinct real roots.
### Step 4: Calculate the Roots Using the Quadratic Formula
The quadratic formula for finding the roots of [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Using [tex]\( a = 1 \)[/tex], [tex]\( b = 42 \)[/tex], and [tex]\( \Delta = 36 \)[/tex]:
[tex]\[ x_1 = \frac{-42 + \sqrt{36}}{2 \cdot 1} \][/tex]
[tex]\[ x_2 = \frac{-42 - \sqrt{36}}{2 \cdot 1} \][/tex]
### Step 5: Simplify the Root Expressions
First, calculate the square root of the discriminant:
[tex]\[ \sqrt{\Delta} = \sqrt{36} = 6 \][/tex]
Now, substitute back into the equations for [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]:
[tex]\[ x_1 = \frac{-42 + 6}{2} = \frac{-36}{2} = -18 \][/tex]
[tex]\[ x_2 = \frac{-42 - 6}{2} = \frac{-48}{2} = -24 \][/tex]
### Solution
The two roots of the quadratic equation [tex]\( a^2 + 42a + 432 = 0 \)[/tex] are:
[tex]\[ a_1 = -18 \][/tex]
[tex]\[ a_2 = -24 \][/tex]
Therefore, the roots of the quadratic equation are [tex]\(-18\)[/tex] and [tex]\(-24\)[/tex].
