While rewriting a quadratic equation into the form ax^2 + bx + c = 0

Question

﻿While rewriting a quadratic equation into the form  ax^2 + bx + c = 0 , John and Katie both made errors. John made an error that resulted in c being incorrect while a and b were correct. Katie made an error that resulted in b being incorrect while a and c were correct. If John obtained 6 and 2 as solutions and Katie obtained -7 and -1 as solutions, and neither of them made any additional errors, what were the correct solutions to the original quadratic equation?

A. -8 and 7
B. -8 and 12
C. -6 and -2
D. 1 and 6
E. 1 and 7

gmatophobia · Accepted Answer

Solution obtained by John

(x-6)(x-2) = 0

x^2 -8x + 12 = 0

We know that the value of +12 is incorrect.

Solution obtained by Katie

(x+7)(x+1) = 0

x^2 + 8x + 7 = 0

From the question stem we know that the value +7x is incorrect.

Combined

x^2 -8x + 7 = 0

x^2 -7x - x + 7 = 0

x(x -7) -1 (x - 7) = 0

(x-1)(x-7) = 0

Therefore the solutions are x = 1 , and x = 7

Option E

JeffTargetTestPrep · Answer

John’s incorrectly written equation was:
(x – 6)(x – 2) = 0
x^2 – 8x + 12 = 0

Since a and b were correct in John’s equation, we have:
a = 1 and b = -8

Katie’s incorrectly written equation was:
(x + 7)(x + 1) = 0
x^2 + 8x + 7 = 0

Since a and c were correct in Katie’s equation, we have:
a = 1 and c = 7

So, in the original equation the coefficients are:
a = 1, b = -8, and c = 7

The correctly written original equation and its solutions are:
x^2 – 8b + 7 = 0
(x – 1)(x – 7) = 0 => x = 1, x = 7

Answer: E

Paras96 · Answer

﻿While rewriting a quadratic equation into the form ax2+bx+c=0, John and Katie both made errors. John made an error that resulted in c being incorrect while a and b were correct. Katie made an error that resulted in b being incorrect while a and c were correct. If John obtained 6 and 2 as solutions and Katie obtained -7 and -1 as solutions, and neither of them made any additional errors, what were the correct solutions to the original quadratic equation?

We know that if ax2+bx+c=0
Product of roots = c/a and sum of roots = -b/a

In the case of John, c being incorrect means the product of roots is incorrect but the sum is correct
Hence the sum of roots for the correct equation is 6+2=8

In the case of Katie, b being incorrect means the sum of roots is incorrect but the product is correct
Hence the product of roots for the correct equation is 7

therefore the correct equation is

=>x^2-8x+7=0
=>x^2-7x-x+7=0
=>x(x-7)-(x-7)=0
=>(x-1)(x-7)=0

hence the root of correct equations are 1 & 7

Hence E

CITY · Answer

Not sure if this is logically correct:

The vertex（middle point) of a Quadratic equation of one variable

equals to

-b/2a，

Since John  get a and b correct , means the  x axis value of John is the same as the corrected function

Meanwhile,  c  means the intercept cordinate of the function to the y axis when x equals to  0 ,
While Kaite  get c correct,  means the  y axis value for vertex（middle point) is the same as the corrected function

So just simply move Katie function drawing to the corrected X axis value,
and simply added number from the middle point: 4+3 =7; 4 -3 = 1

josiewhoasie · Answer

Gmatophobia, your explanation was so helpful! Can you (or someone) please explain to me how you got from 6 and 2 to (x-6)(x-2)? I would have written it as (x+6)(x+2) since the solution was positive (and vice versa for Katie's solutions). Thank you in advance!

Bunuel · Answer

The solutions of (x - 6)(x - 2) = 0 are x = 6 and x = 2. This is because for (x - 6)(x - 2) = 0 to hold, one of the factors must be zero: x - 6 = 0 implies x = 6, and x - 2 = 0 implies x = 2. A similar logic applies to (x + 6)(x + 2) = 0, where the solutions are x = -6 and x = -2 because x + 6 = 0 yields x = -6, and x + 2 = 0 yields x = -2.

Hope it helps.

DanTheGMATMan · Answer

Standard quadratic business:

https://www.youtube.com/watch?v=uj-HcSMkgFg

josiewhoasie · Answer

Thank you both so much! This helped a lot, really appreciate you both.

shyqa · Answer

Is it possible to use the following easy approach? 
(x + q)(x + p) = 0
q * p = c
q + p = b

If Katie's (-7;-1) solution is correct for  c  -> so c = 7
That means ONLY   OR   are the possible solutions

The only answer that has 7 and 1 is E, correct?

GMATinsight · Answer

A SIMPLE APPROACH   
 for any quadratic equation  ax^2 + bx + c = 0 , there are  two roots  α  and    β 
Also, the Sum of the roots,    α  +    β = -b/a  and
Also, the Product of the roots,    α *   β = c/a

Since, John got a and b correct therefore his  sum of the roots (-b/a)  should be correct  (CORRECT)

STOP-1:  P.S. Only  Option E  Satisfies (Sum of roots 8) hence the correct Answer.

Let's proceed anyway

Since, Katie got a and c correct therefore her  Product of the roots (c/a)  should be correct  (CORRECT)

STOP-2:  P.S. Only  Option E  Satisfies (product of roots 7) hence the correct Answer.

Let's proceed anyway

Now, we just have to think of two numbers (roots) whose sum is 8 and the product is 7

Now we know that those two numbers are  7 and 1

Answer: Option E

GMATBusters · Answer

For a quadratic equation, ax^2 + bx + c = 0

Sum of the roots = -b/a

product of the roots = c/a

John has considered correct values of a and b, hence the sum of roots he got must be equal to sum of correct roots = 6+2 = 8

Katie has considered correct values of a and c, hence the product of roots he got must be equal to product of correct roots = (-7)(-1) = 7

From the given options, E : 1, 7 satisfy the condition that sum = 8 and product = 7.

Hence, E is the correct answer.

Abraralam · Answer

Solution obtained by Katie (x+7)(x+1)=0 
should be x^ {2}+8x+7=0 Right ?

sophia1234 · Answer

is there a reason why we don't consider the option that in the quadratic equation, ax^2 + bx + c = 0 'a' could be greater than 1?
 Bunuel

Bunuel · Answer

John’s equation: x^2 - 8x + 12 = 0
Katie’s equation: x ^2 + 8x + 7 = 0

We’re told neither of them made a mistake in the value of  a , and in both equations a = 1. So the original equation must also have a = 1.

KannanAgrawal · Answer

John 
since a and b are correct, sum of roots which is b/a is also correct that means 6+2 is the true  sum of roots,

Katie 
since a and c are correct, the product of roots which is -c/a is correct that means, -7*(-1) is the true  product of roots.

The only option which is directly visible where the sum of roots is 6+2=8 and product of roots is -7*(-1)=7 is option E.

While rewriting a quadratic equation into the form ax^2 + bx + c = 0

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While rewriting a quadratic equation into the form ax^2 + bx + c = 0

Prep Toolkit

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