Factorization Practice Problems - Master Algebraic Techniques

Let's solve the given equation:

$2x^2+4x-6=0$

Let's first simplify the equation, noting that all coefficients as well as the free term are multiples of the number 2, hence we'll divide both sides of the equation by 2:

$2x^2+4x-6=0 \hspace{6pt}\text{/}:2 \\ x^2+2x-3=0\\$Note that the coefficient of the squared term is 1, therefore we can (try to) factor the expression on the left side using quick trinomial factoring:

We'll look for a pair of numbers whose product equals the free term in the expression, and whose sum equals the coefficient of the first-degree term, meaning two numbers $m,\hspace{2pt}n$ that satisfy:

$m\cdot n=-3\\ m+n=2\\$From the first requirement mentioned, that is - from the multiplication, we notice that the product of the numbers we're looking for needs to yield a negative result, therefore we can conclude that the two numbers have different signs, according to multiplication rules. The possible factors of 3 are 3 and 1. Fulfilling the second requirement mentioned, along with the fact that the signs of the numbers we're looking for are equal to each other will lead to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases}m=3 \\ n=-1\end{cases}$

Therefore we'll factor the expression on the left side of the equation to:

$x^2+2x-3=0\\ \downarrow\\ (x+3)(x-1)=0$

From here we'll remember that the result of multiplication between expressions will yield 0 only if at least one of the multiplying expressions equals zero,

Therefore we'll obtain two simple equations and solve them by isolating the unknown on one side:

$x+3=0\\ \boxed{x=-3}$

or:

$x-1=0\\ \boxed{x=1}$

Let's summarize the solution of the equation:

$2x^2+4x-6=0 \\ x^2+2x-3=0 \\ \downarrow\\ (x+3)(x-1)=0 \\ \downarrow\\ x+3=0\rightarrow\boxed{x=-3}\\ x-1=0\rightarrow\boxed{x=1}\\ \downarrow\\ \boxed{x=-3,1}$

Therefore the correct answer is answer B.

To solve this problem, we will factor the given polynomial expression:

Step 1: Identify the greatest common factor (GCF) in the equation $9x^3 - 12x^2 = 0$. The GCF of the terms $9x^3$ and $12x^2$ is $3x^2$.

Step 2: Factor out the GCF from the polynomial:

$9x^3 - 12x^2 = 3x^2(3x - 4) = 0$.

Step 3: Apply the zero-product property. Set each factor equal to zero:

• $3x^2 = 0$
• $3x - 4 = 0$

Step 4: Solve each equation for $x$:

For $3x^2 = 0$, divide by 3:

$x^2 = 0$ → $x = 0$.

For $3x - 4 = 0$, add 4 to both sides and then divide by 3:

$3x = 4$
$x = \frac{4}{3}$.

Thus, the solutions to the equation $9x^3 - 12x^2 = 0$ are $x = 0$ and $x = \frac{4}{3}$.

Therefore, the correct answer is:

$x=0, x=\frac{4}{3}$

To solve this problem, we'll follow these steps:

• Step 1: Factor the equation $x^2 + x = 0$.
• Step 2: Use the Zero-Product Property to solve for $x$.

Now, let's work through each step:

Step 1: Start by factoring the left-hand side of the equation:
$x^2 + x = x(x + 1)$

Step 2: Apply the Zero-Product Property:
Since $x(x + 1) = 0$, we have two possible equations:
1) $x = 0$
2) $x + 1 = 0$

For the second equation, solve for $x$:
$x + 1 = 0$ implies $x = -1$

Therefore, the solutions to the equation are $x = 0$ and $x = -1$.

Hence, the value of the parameter $x$ is $x = 0, x = -1$.

To solve the equation $-9x + 3x^2 = 0$, follow these steps:

Step 1: Factor the equation.

Observe that both terms in the equation share a common factor, $3x$. We can factor this out:

$-9x + 3x^2 = 3x(-3 + x)$.

The factored equation is $3x(-3 + x) = 0$.

Step 2: Apply the zero product property.

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two equations to solve:

• $3x = 0$
• $-3 + x = 0$

Step 3: Solve each equation.

• For $3x = 0$, divide both sides by 3 to solve for $x$:
• $x = 0$
• For $-3 + x = 0$, add 3 to both sides to solve for $x$:
• $x = 3$

Therefore, the solutions to the equation $-9x + 3x^2 = 0$ are $x = 0$ and $x = 3$.

Matching these solutions to the given choices, the correct answer is choice 3: $x = 0, x = 3$.

Thus, the values of $x$ that satisfy the equation are $x = 0$ and $x = 3$.

Let's solve the given equation:

$x^2-1=0$ We will do this simply by isolating the unknown on one side and taking the square root of both sides:

$x^2-1=0 \\ x^2=1\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\\ \boxed{x=\pm1}$

Therefore, the correct answer is answer A.