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655-705 (Hard)|   Algebra|      
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Bunuel
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By what factor should the smaller root of the equation 6x^2 -13x + 6 = 21 Oct 2021, 01:30
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65% (hard)

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59% (02:01) correct 41% (02:01) wrong based on 285 sessions

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By what factor should the smaller root of the equation \(6x^2 -13x + 6 = 0\) increase so that it equals larger root of the equation.

A. 4/9
B. 5/6
C. 6/5
D. 2
E. 9/4
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E
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By what factor should the smaller root of the equation 6x^2 -13x + 6 = 21 Oct 2021, 05:18
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= 6X^2 - 13X + 6
= 6X^2 - 9X - 4X + 6
= (6X - 9)(6X - 4)
= X { 3/2 , 2/3}

Smaller root = 2/3
Smaller root multiplied by (2/3) * (9/4) = (3/2) larger root

Correct Option E

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By what factor should the smaller root of the equation 6x^2 -13x + 6 = 21 Oct 2021, 05:32
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6x^2−13x+6=0
6x^2 - 9x - 4x + 6 =0
= 3x (2x -3) - 2(2x-3)=0
=(3x-2) (2x-3)
=3x = 2
x=2/3

2x=3
x=3/2

Smaller root = 2/3
Large root = 3/2
Factor the smaller root needs to be increased to be equal to larger root = 3/2 *3/2 =9/4

IMO E
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By what factor should the smaller root of the equation 6x^2 -13x + 6 = 21 Oct 2021, 05:58
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Converting equation to standard Quadratic, sum of roots = 13/6, product of roots = 1.

Since determinant is >0, plugging simple reciprocal numbers gives 2/3 & 3/2 adding upto 13/6

Answer will essentially be (larger root)^2.

Hence, 3/2*3/2= 9/4

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By what factor should the smaller root of the equation 6x^2 -13x + 6 = 03 Nov 2021, 07:50
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Bunuel
By what factor should the smaller root of the equation \(6x^2 -13x + 6 = 0\) increase so that it equals larger root of the equation.

A. 4/9
B. 5/6
C. 6/5
D. 2
E. 9/4
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This question is a part of Are You Up For the Challenge: 700 Level Questions collection.­
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By what factor should the smaller root of the equation 6x^2 -13x + 6 = 13 Aug 2024, 05:13
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Bunuel
By what factor should the smaller root of the equation \(6x^2 -13x + 6 = 0\) increase so that it equals larger root of the equation.

A. 4/9
B. 5/6
C. 6/5
D. 2
E. 9/4
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­Bunuel egmat why do we use multiplication to increase it and not addition?
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By what factor should the smaller root of the equation 6x^2 -13x + 6 = 13 Aug 2024, 07:44
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­Great Question! The key is to understand what is being asked when we say "factor"?

What do we mean when we say "by factor"?

When we use the term "factor" in mathematics, we're referring to a multiplicative relationship. It implies that one quantity is being scaled up or down by multiplying it by a certain number. This is different from an additive relationship where we're increasing or decreasing a quantity by adding or subtracting a fixed amount.

Examples of factor relationships:
a) If a company's revenue grows "by a factor of 2", it means the new revenue is double the original (Revenue × 2).
b) When we say "the image is scaled by a factor of 0.5", it means the new size is half the original (Size × 0.5).
c) If a population increases "by a factor of 1.5 over a decade", it means the new population is 1.5 times the original (Population × 1.5).

How the question would be worded for addition:
If the test maker wanted to imply addition instead of multiplication, the question could be worded as:

"What amount should be added to the smaller root of the equation 6x^2 - 13x + 6 = 0 so that it equals the larger root of the equation?"

Examples of addition-based questions:
a) "How much must be added to 5 to obtain 12?"
b) "By what amount does the larger root exceed the smaller root in the equation x^2 - 5x + 6 = 0?"
c) "What is the difference between the larger and smaller roots of the equation 2x^2 - 7x + 3 = 0?"

I hope this helps!

-Rajat­
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By what factor should the smaller root of the equation 6x^2 -13x + 6 = 08 Oct 2025, 13:04
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good question, a tricky one, my first answer was 5/6 but now i understand that the main part is understanding the term "factor"
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