What are the solutions of the equation [tex]\(4x^2 + 3x = 24 - x\)[/tex]?

A. [tex]\(-3, 2\)[/tex]
B. [tex]\(-3\)[/tex] or [tex]\(2\)[/tex]
C. [tex]\(-2, 3\)[/tex]
D. [tex]\(-2\)[/tex] or [tex]\(3\)[/tex]



Answer :

Let's solve the equation [tex]\(4x^2 + 3x = 24 - x\)[/tex] step by step.

1. First, bring all terms to one side to set the equation to zero:
[tex]\[ 4x^2 + 3x - 24 + x = 0 \][/tex]
Combine like terms:
[tex]\[ 4x^2 + 4x - 24 = 0 \][/tex]

2. Simplify the equation by dividing all terms by 4:
[tex]\[ x^2 + x - 6 = 0 \][/tex]

3. Factor the quadratic equation [tex]\(x^2 + x - 6\)[/tex]:
We need to find two numbers that multiply to [tex]\(-6\)[/tex] and add to [tex]\(1\)[/tex]. These numbers are [tex]\(3\)[/tex] and [tex]\(-2\)[/tex]:
[tex]\[ (x + 3)(x - 2) = 0 \][/tex]

4. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ x + 3 = 0 \quad \text{or} \quad x - 2 = 0 \][/tex]
Solving each equation:
[tex]\[ x = -3 \quad \text{or} \quad x = 2 \][/tex]

Therefore, the solutions to the equation [tex]\(4x^2 + 3x = 24 - x\)[/tex] are [tex]\(\boxed{-3\text{ or }2}\)[/tex].