GMAT Quant | Quadratic Equations #7

This GMAT Practice Question in quant is a problem solving question in Quadratic Equations in Algebra. Concept tested: Condition when the roots or solutions of a quadratic equation will be real and distinct. A GMAT sample question to consolidate concepts relating to nature of roots of a quadratic equation. A 650 level GMAT question combining concepts in quadratic equations and basic inequalities.

Question 7: What is the highest integral value of 'k' for which the quadratic equation x² - 6x + k = 0 will have two real and distinct roots?

1. 9
2. 7
3. 3
4. 8
5. 12

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Video Explanation

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Explanatory Answer

Step 1 of solving this GMAT Quadratic Equations Question: Nature of Roots of Quadratic Equation Theory

D is called the Discriminant of a quadratic equation.
D = b² - 4ac for quadratic equations of the form ax² + bx + c = 0.
If D > 0, roots of such quadratic equations are Real and Distinct
If D = 0, roots of such quadratic equations are Real and Equal
If D < 0, roots of such quadratic equations are Imaginary. i.e., such quadratic equations will NOT have real roots.
In this GMAT sample question in algebra, we have to determine the largest integer value that ‘k’ can take such that the discriminant of the quadratic equation is positive.

Step 2 of solving this GMAT Quadratic Equations Question: Compute maximum value of k that will make the discriminant of the quadratic equation positive

The given quadratic equation is x² - 6x + k = 0.
In this equation, a = 1, b = -6 and c = k
The value of the discriminant, D = 6² - 4 * 1 * k
We have compute values of 'k' for which D > 0.
i.e., 36 – 4k > 0
Or 36 > 4k or k < 9.

The question is "What is the highest integer value that k can take?"
If k < 9, the highest integer value that k can take is 8.

Choice D is the correct answer.