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What is the completely factored form of $6 x^2-13 x-5$?


A$(2 x-5)(3 x+1)$
B$(2 x+5)(3 x-1)$
C$(2 x-1)(3 x-5)$
D$(2 x+1)(3 x+5)$

Verified Answer :

  • Identify the coefficients \(a=6\), \(b=-13\), and \(c=-5\) of the quadratic expression.
  • Find two numbers that multiply to \(a \times c = -30\) and sum to \(b = -13\), which are \(-15\) and \(2\).
  • Rewrite the expression as \(6x^2 - 15x + 2x - 5\) and factor by grouping to obtain the final form.
  • The correct factored form is \(\boxed{(2x - 5)(3x + 1)}\).

Explanation

  1. Analysis of the Quadratic Expression

We are given the quadratic expression \(6x^2 - 13x - 5\) and we want to find its completely factored form. A quadratic expression in the standard form \(ax^2 + bx + c\) can be factored by finding two numbers that multiply to the product \(a \times c\) and add up to \(b\). Here, \(a = 6\), \(b = -13\), and \(c = -5\). Thus, we need to find two numbers that multiply to \(6 \times (-5) = -30\) and add up to \(-13\).

  1. Factoring by Grouping

We look for two numbers whose product is \(-30\) and whose sum is \(-13\). Testing pairs of factors of \(-30\):

  • The pair \(-15\) and \(2\) satisfies both conditions: \(-15 \times 2 = -30\) and \(-15 + 2 = -13\).

Now, we rewrite the middle term \(-13x\) using these two numbers:
\[6x^2 - 15x + 2x - 5\]
Next, we group the terms:
\[(6x^2 - 15x) + (2x - 5)\]
Extracting the common factors from each group:

  • From the first group \((6x^2 - 15x)\), we can factor out \(3x\), giving \(3x(2x - 5)\).
  • From the second group \((2x - 5)\), we can factor out \(1\), giving \(1(2x - 5)\).

So the expression becomes:
\[3x(2x - 5) + 1(2x - 5)\]
Factoring out the common binomial \((2x - 5)\), we get:
\[(2x - 5)(3x + 1)\]

  1. Verification and Conclusion

Finally, we confirm the factorization by expanding the result:
\[(2x - 5)(3x + 1) = 2x(3x) + 2x(1) - 5(3x) - 5(1) = 6x^2 + 2x - 15x - 5 = 6x^2 - 13x - 5\]
This matches our original expression. Comparing this to the given options, we identify the correct factored form as \((2x - 5)(3x + 1)\). The final answer is \(\boxed{(2x - 5)(3x + 1)}\).

Examples

Factoring quadratics is a fundamental skill in algebra with many real-life applications. For example, engineers use it to model the trajectory of a projectile, where the expression represents the height of an object at a given time. By finding the factored form, they can easily determine when the object hits the ground (by setting the expression equal to zero). Similarly, in economics, businesses use it to find the break-even point for products by factoring profit functions. Mastering this skill allows for solving complex real-world problems involving relationships between variables.

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