Factor The Following Expression:$\[ 6x^2 - 7x + 2 \\]Fill In The Blanks In The Factorization:$\[ (2x - [?])(3x - [?]) \\]

Feb 26, 2025 by ADMIN 122 views

Factor the Following Expression: A Step-by-Step Guide

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Introduction

In algebra, factoring is a process of expressing a polynomial as a product of simpler polynomials. It is an essential skill in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on factoring a quadratic expression, which is a polynomial of degree two. We will use the given expression $6x^2 - 7x + 2$ as an example and guide you through the process of factoring it.

Understanding the Factorization

The given expression is a quadratic expression, and we are asked to factor it in the form $(2x - a)(3x - b)$ . To do this, we need to find the values of $a$ and $b$ that satisfy the equation. We can start by expanding the product $(2x - a)(3x - b)$ using the distributive property:

(2x - a)(3x - b) = 6x^2 - 2ax - 3bx + ab

We can see that the expanded product has three terms: $6x^2$ , $-2ax$ , and $-3bx$ . We also have the constant term $ab$ . We can equate the coefficients of the terms in the expanded product with the corresponding terms in the given expression:

6x^2 - 2ax - 3bx + ab = 6x^2 - 7x + 2

Equating Coefficients

We can equate the coefficients of the $x^2$ terms, the $x$ terms, and the constant terms separately:

Coefficients of $x^2$ : $6 = 6$
Coefficients of $x$ : $-2a - 3b = -7$
Constant terms: $ab = 2$

Solving the System of Equations

We have a system of two equations with two variables:

-2a - 3b = -7

ab = 2

We can solve this system of equations using substitution or elimination. Let's use substitution. We can solve the second equation for $a$ in terms of $b$ :

a = \frac{2}{b}

Substituting this expression for $a$ into the first equation, we get:

-2\left(\frac{2}{b}\right) - 3b = -7

Simplifying this equation, we get:

-\frac{4}{b} - 3b = -7

Multiplying both sides by $b$ , we get:

-4 - 3b^2 = -7b

Rearranging this equation, we get:

3b^2 - 7b + 4 = 0

Solving the Quadratic Equation

We have a quadratic equation in the form $ax^2 + bx + c = 0$ . We can solve this equation using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = 3$ , $b = -7$ , and $c = 4$ . Plugging these values into the quadratic formula, we get:

b = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(3)(4)}}{2(3)}

Simplifying this expression, we get:

b = \frac{7 \pm \sqrt{49 - 48}}{6}

b = \frac{7 \pm \sqrt{1}}{6}

b = \frac{7 \pm 1}{6}

We have two possible values for $b$ :

b = \frac{7 + 1}{6} = \frac{8}{6} = \frac{4}{3}

b = \frac{7 - 1}{6} = \frac{6}{6} = 1

Finding the Values of $a$

We can find the values of $a$ by substituting the values of $b$ into the equation $a = \frac{2}{b}$ :

If $b = \frac{4}{3}$ , then $a = \frac{2}{\frac{4}{3}} = \frac{6}{4} = \frac{3}{2}$
If $b = 1$ , then $a = \frac{2}{1} = 2$

Writing the Factored Form

We can write the factored form of the given expression using the values of $a$ and $b$ :

If $a = \frac{3}{2}$ and $b = \frac{4}{3}$ , then the factored form is $(2x - \frac{3}{2})(3x - \frac{4}{3})$
If $a = 2$ and $b = 1$ , then the factored form is $(2x - 2)(3x - 1)$

Conclusion

In this article, we have factored the quadratic expression $6x^2 - 7x + 2$ in the form $(2x - a)(3x - b)$ . We have found the values of $a$ and $b$ by solving a system of equations and have written the factored form of the expression using these values. We have also discussed the importance of factoring in mathematics and its applications in various fields.

Final Answer

The final answer is: