# GMAT Math Questions | Quadratic Equations #13

#### GMAT Sample Questions | Quadratic Algebra Questions

This GMAT Quant Practice Question is a problem solving question in Quadratic Equations in Algebra. Concept: Sum of roots of quadratic equations and product of roots of quadratic equations and number properties basics and counting methods basics. A medium difficulty, 650 level GMAT practice question in algebra.

Question 13 : If p > 0, and x2 - 11x + p = 0 has integer roots, how many integer values can 'p' take?

1. 6
2. 11
3. 5
4. 10
5. Infinitely many

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#### Step 1 of solving this GMAT Quadratic Equations Question : Compute sum and product of roots of this quadratic equation

The given quadratic equation is x2 - 11x + p = 0.
The sum of the roots of this quadratic equation = $-$frac{b}{a}$ = -$\frac{-11}{1}$ = 11. The product of the roots of this quadratic equation = $\frac{c}{a}$ = $\frac{p}{1}$ = p. #### Step 2 of solving this GMAT Quadratic Algebra Question : Deduce properties about roots of this quadratic equation Inference 1: The question states that p > 0. 'p' is the product of the roots of this quadratic equation. So, the product of the two roots is positive. The product of two numbers is positive if either both the numbers are positive or both the numbers are negative. Inference 2: We also know that the sum of the roots = 11, which is positive. The sum of two negative numbers cannot be positive. So, both the roots have to be positive. #### Step 3 of solving this GMAT Quadratic Algebra Problem : List values possible for roots We also know that the roots are integers. So, we have to find different ways of expressing 11 as a sum of two positive integers. Possibility 1:$1, 10)
Possibility 2: (2, 9)
Possibility 3: (3, 8)
Possibility 4: (4, 7)
Possibility 5: (5, 6)

Each of these pairs, will result in a different value for p.
So, p can take 5 different values.

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