Certainly! Let's solve the given quadratic equation step-by-step.
The quadratic equation given is:
[tex]\[
(4x + 3)(x - 6) = 0
\][/tex]
### Step 1: Expand the equation
First, expand the left-hand side by using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(4x + 3)(x - 6)
\][/tex]
Multiply each term in the first binomial by each term in the second binomial:
[tex]\[
4x \cdot x + 4x \cdot (-6) + 3 \cdot x + 3 \cdot (-6)
\][/tex]
Simplify each of the products:
[tex]\[
4x^2 - 24x + 3x - 18
\][/tex]
Combine like terms:
[tex]\[
4x^2 - 21x - 18
\][/tex]
So our quadratic equation is now:
[tex]\[
4x^2 - 21x - 18 = 0
\][/tex]
### Step 2: Solve the quadratic equation
To solve the quadratic equation [tex]\( 4x^2 - 21x - 18 = 0 \)[/tex], we use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 4 \)[/tex], [tex]\( b = -21 \)[/tex], and [tex]\( c = -18 \)[/tex].
### Step 3: Calculate the discriminant
The discriminant [tex]\( \Delta \)[/tex] is:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (-21)^2 - 4 \cdot 4 \cdot (-18)
\][/tex]
Calculate the discriminant:
[tex]\[
\Delta = 441 + 288 = 729
\][/tex]
Since the discriminant is positive, there are two distinct real roots.
### Step 4: Find the roots using the quadratic formula
Substitute [tex]\( \Delta \)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{-(-21) \pm \sqrt{729}}{2 \cdot 4}
\][/tex]
This simplifies to:
[tex]\[
x = \frac{21 \pm 27}{8}
\][/tex]
### Step 5: Calculate the two roots
1. For the positive case [tex]\( 21 + 27 \)[/tex]:
[tex]\[
x = \frac{48}{8} = 6
\][/tex]
2. For the negative case [tex]\( 21 - 27 \)[/tex]:
[tex]\[
x = \frac{-6}{8} = -\frac{3}{4}
\][/tex]
### Step 6: State the solution
The solutions to the quadratic equation [tex]\( (4x + 3)(x - 6) = 0 \)[/tex] are:
[tex]\[
x = 6 \quad \text{and} \quad x = -\frac{3}{4}
\][/tex]
Therefore, the correct answer is:
[tex]\[
x = \left\{\boxed{6}, \boxed{-\frac{3}{4}} \right\}
\][/tex]
