What are the roots of the quadratic equation x^2 + bx + c = 0 if the

Question

What are the roots of the quadratic equation x^2 + bx + c = 0 if the roots are distinct and at equal distance from 5 on the number line?

(1) The product of the roots of the equation x^2 + bx + c = 0 is 21

(2) x – 7 is a factor of the expression x^2 + bx + c

LeoN88 · Answer

D. Each independently sufficient.

1. x1 * x2 = 21 Possible: (21,1) (3,7). Only (3,7) are distinct roots equidistant from 5.
Alternatively solve (5+x)(5-x)=21

2. If x1= 7, x2 can be 7 or 3. Only (3,7) are distinct roots equidistant from 5.

CrackverbalGMAT · Answer

Although this question can be solved algebraically using Quadratic equations concepts, graphing the equation using the question data will provide more clarity.

Let the roots of the equation \(x^2\) + bx + c = 0 be, p and q respectively. Then, we know that pq = c and p+q = b. Also since the roots are distinct and at equal distance from 5 on the number line, they should look like this:

Attachment:

12th June 2019 - Reply 6.JPG [ 19.1 KiB | Viewed 9289 times ]

From the graph, it’s easy to conclude that 5 – p = q – 5. Simplifying this, we get, p + q = 10. Therefore, b = 10. So, we already know the sum of the roots of the equation.

This means, the equation now becomes \(x^2\) + 10x + c = 0.
If we can get information about the product of the roots, we will able to solve the equation and find the roots.

From statement I, we have the information about the product of the roots of the equation i.e. c = 21. Therefore, the equation becomes \(x^2\) + 10x + 21 = 0. This is a unique equation and we can find out the roots. So, statement I alone is sufficient. Possible answer options are A or D; answer options B, C and E can be ruled out.

From statement 2, we know that x – 7 is a factor of the expression \(x^2\) + 10x + c.

If x – a is a factor of f(x), where f(x) is a polynomial expression, then x = a will be a root of f(a) = 0.

This means x=7 is a root of the equation \(x^2\) + 10x + c = 0. Since the roots of this equation are equidistant from 5, the other root has to be 3. Therefore, this data is sufficient.
The correct answer option is D.

Hope this helps!

Kritisood · Answer

for the quad equation x^2 + bx + c = 0 let the distance of the root of the equation from 5 be d. Hence, the two roots are (5-d) and (5+d)
the sum of the roots = 10
therefore, -b/a=10
a is 1 (coefficient of x)
-b=10
b=10

statement 1 product of the roots is 21
c/a =21
c=21
x^2-10x+21=0 we can solve and find the value of the roots

statement 2 
x-7 is a factor, therefore, x=7 satisfies the equation and is one of the roots
5 + d = 7 so d = 2 and hence the other root is 5 – d = 3. or
5 – d = 7 so, d = -2 and hence the other root is 5 +(-2) = 3

In both cases, we get the same values of the roots.

sufficient

bumpbot · Answer

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