GMAT Quant | Quadratic Equations #8

This GMAT sample question is a math problem solving question from Quadratic Equations. Concept tested: Coefficients of quadratic expressions and their relation to the sum and product of roots of quadratic equations. A sub 600 level GMAT practice question in algebra.

Question 8: If one of the roots of the quadratic equation x² + mx + 24 = 0 is 1.5, then what is the value of m?

1. -22.5
2. 16
3. -10.5
4. -17.5
5. Cannot be determined

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Video Explanation

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Explanatory Answer

Approach to solve this GMAT Quadratic Equations Question: If 1.5 is a root of the quadratic equation, substituting x = 1.5 in the equation will satisfy the equation.

The given quadratic equation is x² + mx + 24 = 0
Substitute x = 1.5 in the above equation because 1.5 is a root of the equation.
(1.5)² + 1.5m + 24 = 0
2.25 + 1.5m + 24 = 0
1.5m = -26.25 Or m = \\frac{-26.25}{1.5}) = -17.5

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Alternative Method

Step 1 of solving this GMAT Quadratic Equations Question: Sum and Product of Roots of Quadratic Equations Theory

For quadratic equations of the form ax² + bx + c = 0, whose roots are α and β,
Sum of the roots, α + β = \-\frac {b} {a}), and product of the roots, αβ = \\frac {c} {a}).
From the question stem, we know that one of the roots is 1.5. Let α be 1.5.

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Step 2 of solving this GMAT Quadratic Equations Question: Compute the second root of the equation

Product of the roots of the quadratic equation x² + mx + 24 = 0 is \\frac {c} {a}) = \(\frac {24} {1})) = 24.
i.e., α * β = 24 where α is 1.5.
1.5 * β = 24
β = \(\frac {24} {1.5}))
β = 16

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Step 3 of solving this GMAT Quadratic Equations Question: Compute the value of ‘m’

In the given equation, m is the co-efficient of the x term.
We know that the sum of the roots of quadratic equations of the form ax² + bx + c = 0 is \\frac {-b} {a})=\\frac {-m} {1}) = -m
Sum of the roots = 16 + 1.5 = 17.5
Sum of the roots = -m
If –m = 17.5, the value of m = -17.5

Choice D is the correct answer.