Question

Which of the following is a correct factorization of \(-12 x^{2}+147 ?\) (A) \(-3(2 x+7)^{2}\) (B) \(3(2 x-7)(2 x+7)\) (C) \(-2(2 x-7)(2 x+7)\) (D) \(-3(2 x-7)(2 x+7)\)

Short answer

The correct factorization of \(-12 x^{2}+147\) is given by Option (D) \(-3(2 x-7)(2 x+7)\).

Step-by-step solution

Factorize given options

Begin by expanding all the options, which allows us to check which option is equal to the original expression. Arrange the expressions in the form \(ax^{2} + bx + c\), where a, b, and c are constants. This can be performed using the FOIL method or the distributive property.

Verify Option (A)

Expand Option (A) \(-3(2 x+7)^{2}\). This gives us \(-12 x^{2} - 42x + 147\), which is not equal to the original equation \(-12 x^{2}+147\). So, Option (A) is incorrect.

Verify Option (B)

Expand Option (B) \(3(2 x-7)(2 x+7)\). This gives us \(12 x^{2} - 147\), which is not equal to the original equation \(-12 x^{2}+147\). So, Option (B) is incorrect.

Verify Option (C)

Expand Option (C) \(-2(2 x-7)(2 x+7)\). This gives us \(-8 x^{2} + 98\), which is not equal to the original equation \(-12 x^{2}+147\). Therefore, Option (C) is incorrect.

Verify Option (D)

Expand Option (D) \(-3(2 x-7)(2 x+7)\). This gives \(-12 x^{2} + 147\), which is indeed equal to the original equation \(-12 x^{2}+147\). Therefore, Option (D) is correct.

Key concepts

FOIL Method

Understanding the FOIL method is crucial when dealing with algebra, especially when you're asked to multiply two binomials. FOIL stands for First, Outer, Inner, Last -- essentially the order in which you multiply the terms of each binomial.

For example, if you have \( (a + b)(c + d) \), you would multiply them in this order: First \( a \times c \), Outer \( a \times d \), Inner \( b \times c \), and Last \( b \times d \). This method ensures that you account for all possible multiplications. To put it plainly, it's like making sure you shake hands with everyone at a gathering -- you wouldn't want to miss tapping elbows with anyone!

Once terms are multiplied, they can be summed up or simplified as needed to get the expanded form of the expression. Typically, you’d then collect like terms to simplify the result as much as possible.

Distributive Property

The distributive property, also known as the distributive law of multiplication, is a way to multiply a sum by distributing the multiplier to each term in the parentheses. The property is relevant for expansion and factorization problems alike.

An example of using this property would be expanding \( 3(x + 4) \). According to the distributive property, you'd multiply 3 by \( x \) and then by 4, resulting in \( 3x + 12 \). This property is a fundamental principle in algebra that paves the way to simplify expressions and solve equations.

When recognizing the use of the distributive property in our exercise, it becomes clear how the options were expanded. Understanding this property aids in verifying each option against the original expression. It's like handing out slices of a pie; each slice represents a part of the whole, and in the end, they all add up to the entire pie.

Quadratic Equations

Quadratic equations come into play when you’re dealing with polynomial expressions of second degree, typically in the form \( ax^{2} + bx + c = 0 \), where \( a \) is not equal to zero. These are fundamental in various areas of mathematics and can model numerous real-life scenarios.

Factoring is a key strategy in solving quadratic equations, which involves breaking down the equation into a product of simpler expressions that can be set to zero to find the solution. The correct factorization essentially unlocks the values of \( x \) that satisfy the equation.

In the context of our exercise, determining the correct factorization is akin to solving a puzzle; each piece must accurately fit with one another to see the complete picture. Option (D), \( -3(2x-7)(2x+7) \), matches our original expression, meaning we found a set of factors that combine to form the desired quadratic equation.