If 5 is one solution of the equation x^2 + 4x + m = 5, where m is a co

Official Explanation

If 5 is a solution, then we can plug it in and find m:

25 + 20 + m = 5

Therefore, m = -40 and the equation is:

x^2 + 4x - 40 = 5

x^2 + 4x - 45 = 0

On the GMAT, when we get a quadratic equation, it can usually be solved by factoring. In this case:

(x + ?)(x + ?) = 0

The two question marks must add to +4 and multiply to -45, so we have

(x + 9)(x - 5) = 0

The two solutions are x = -9 and x = 5, and we already had 5.

The correct answer is (A).

Video Explanation

This is a fairly simple question on Quadratic equations. The concept being tested here is the sum of the roots of a quadratic equation.

For a standard quadratic equation of the form a\(x^2\) + bx + c =0,

Sum of roots = -\(\frac{b}{a}\)

Product of roots = \(c/a\)

The equation given in the question can be rewritten so that it can be compared with the standard form. When we re-write the equation, we have,

\(x^2\) + 4x + (m-5) = 0. Comparing this with the standard form, we have,

a = 1, b = 4 and c = (m-5).
We also know that one of the roots is 5, say α; we are trying to find the other root, say β.

Sum of the roots = α + β = - \(\frac{b}{a}\) = -4.

Substituting the value of α in the above equation and simplifying, we have β = -9. So, the other solution is -9.
The correct answer option is A.

Note that you do not have to solve the quadratic equation to find out the other root, as long as you know the concepts of the sum and product of roots of a quadratic equation.

Hope that helps!

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