Solve Using the Quadratic Formula x^3-3x^2-4x+12
The problem presented here is a mathematical one, requiring the use of the quadratic formula to solve a polynomial equation. However, the stated equation is not a quadratic equation but a cubic one, given by x^3 - 3x^2 - 4x + 12 = 0. The quadratic formula is applicable for equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The cubic equation involves an x^3 term (a third-degree term) and therefore cannot be solved using the quadratic formula. Instead, one would need to use methods suitable for solving cubic equations, such as factorization, the Rational Root Theorem, synthetic division, or numerical methods if an analytical solution is difficult to obtain.
$x^{3} - 3 x^{2} - 4 x + 12$
Answer
Solution:
Step 1:
Begin by equating the cubic polynomial to zero: $x^3 - 3x^2 - 4x + 12 = 0$.
Step 2:
Commence factoring the equation.
Step 2.1:
Separate the terms into two groups for easier factoring: $(x^3 - 3x^2) - (4x - 12) = 0$.
Step 2.1.1:
Extract the highest common factor from each pair of terms: $x^2(x - 3) - 4(x - 3) = 0$.
Step 2.2:
Identify and factor out the common binomial factor from the entire expression: $(x - 3)(x^2 - 4) = 0$.
Step 2.3:
Express the constant term in the second factor as a square: $(x - 3)(x^2 - 2^2) = 0$.
Step 2.4:
Apply the difference of squares rule to factor further.
Step 2.4.1:
Employ the difference of squares formula: $(x - 3)((x + 2)(x - 2)) = 0$.
Step 2.4.2:
Simplify the expression by removing redundant parentheses: $(x - 3)(x + 2)(x - 2) = 0$.
Step 3:
Realize that if any factor equals zero, the entire expression is zero.
Step 4:
Solve for $x$ by setting the first factor equal to zero.
Step 4.1:
Set the first factor to zero: $x - 3 = 0$.
Step 4.2:
Isolate $x$ by adding $3$ to both sides: $x = 3$.
Step 5:
Solve for $x$ by setting the second factor equal to zero.
Step 5.1:
Set the second factor to zero: $x + 2 = 0$.
Step 5.2:
Isolate $x$ by subtracting $2$ from both sides: $x = -2$.
Step 6:
Solve for $x$ by setting the third factor equal to zero.
Step 6.1:
Set the third factor to zero: $x - 2 = 0$.
Step 6.2:
Isolate $x$ by adding $2$ to both sides: $x = 2$.
Step 7:
Combine the solutions to find the roots of the original equation: $x = 3, -2, 2$.
Knowledge Notes:
The problem involves solving a cubic polynomial equation. The solution process uses factoring techniques, which are fundamental in algebra. The steps include:
Setting the polynomial equal to zero.
Grouping terms and factoring by grouping.
Factoring out the greatest common factor (GCF).
Recognizing and applying the difference of squares formula, which states that $a^2 - b^2 = (a + b)(a - b)$.
Setting each factor equal to zero to solve for the variable $x$.
The solutions to the equation are the values of $x$ that satisfy the original equation.
This process relies on the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero. This property allows us to set each factor equal to zero and solve for the roots of the equation. The difference of squares is a specific case of factoring where two terms are perfect squares separated by a subtraction sign, allowing them to be factored into the product of a sum and difference.
