Solve by Factoring 6x^2+96x=-384

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:

1. Factoring Quadratic Equations: To solve a quadratic equation by factoring, one must express the equation in the form $a{x}^2+bx+c=0$ and then factor it into a product of binomials or other expressions that can be set to zero.

2. Common Factor: Factoring out the greatest common factor (GCF) simplifies the quadratic expression and prepares it for further factoring if possible.

3. 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.

4. 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.

5. 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.

6. 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.