Solve by Factoring 6x^2+96x=-384
The question provided requires you to solve a quadratic equation by the method of factoring. Factoring is a process that involves rewriting the equation in a form such that the left-hand side becomes a product of two linear expressions set equal to zero. You're expected to manipulate the given quadratic equation, which currently has a leading coefficient of 6, a linear coefficient of 96, and a constant term moved to the right-hand side of the equation to produce -384. You will adjust it into a standard form (ax^2 + bx + c = 0), factor the quadratic expression, and then use the property that if the product of two expressions is zero, then at least one of the expressions must be zero. This will enable you to find the values of x that satisfy the equation.
$6 x^{2} + 96 x = - 384$
Answer
Solution:
Step 1:
Move $384$ to the left side to form a standard quadratic equation: $6x^2 + 96x + 384 = 0$.
Step 2:
Extract the common factor of $6$ from each term:
From $6x^2$, we get $6(x^2)$.
From $96x$, we get $6(16x)$.
From $384$, we get $6(64)$.
Thus, the equation becomes $6(x^2 + 16x + 64) = 0$.
Step 3:
Identify the perfect square trinomial within the parentheses:
Recognize that $64$ is $8^2$.
Confirm that $16x$ is the double product of $x$ and $8$, i.e., $16x = 2 \cdot x \cdot 8$.
Rewrite the trinomial as $6(x^2 + 2 \cdot x \cdot 8 + 8^2) = 0$.
Factor the trinomial using the formula $(a + b)^2 = a^2 + 2ab + b^2$, where $a = x$ and $b = 8$, to get $6(x + 8)^2 = 0$.
Step 4:
Divide the equation by $6$ to isolate the perfect square:
Divide $6(x + 8)^2$ by $6$ to get $(x + 8)^2 = \frac{0}{6}$.
Simplify the equation to $(x + 8)^2 = 0$.
Step 5:
Solve for $x$ by setting the expression inside the square to zero: $x + 8 = 0$.
Step 6:
Isolate $x$ by subtracting $8$ from both sides: $x = -8$.
Knowledge Notes:
Factoring Quadratic Equations: To solve a quadratic equation by factoring, one must express the equation in the form $ax^2 + bx + c = 0$ and then factor it into a product of binomials or other expressions that can be set to zero.
Common Factor: Factoring out the greatest common factor (GCF) simplifies the quadratic expression and prepares it for further factoring if possible.
Perfect Square Trinomial: A perfect square trinomial is an expression of the form $a^2 + 2ab + b^2$, which factors into $(a + b)^2$. Recognizing this pattern is crucial for factoring certain quadratic equations.
Zero Product Property: If a product of factors equals zero, at least one of the factors must be zero. This property allows us to set each factor of a factored quadratic equation to zero and solve for the variable.
Simplifying Equations: When both sides of an equation are divisible by the same non-zero number, we can divide both sides by that number to simplify the equation.
Solving Simple Equations: To solve for a variable, we perform inverse operations to isolate the variable on one side of the equation. This often involves adding or subtracting terms and dividing or multiplying both sides by a number.
