Quadratic Equations & Parabola #15

This GMAT sample question is a problem solving question in Quadratic Equations in Algebra. Concept tested: Relating coefficients of quadratic equations to points on the curve described by the equation - elementary coordinate geometry. A medium difficulty, 650 to 700 level GMAT maths question in algebra.

Question 15 : If the curve described by the equation y = x² + bx + c cuts the x-axis at -4 and y axis at 4, at which other point does it cut the x-axis?

1. -1
2. 4
3. 1
4. -4
5. 0

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Video Explanation

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Explanatory Answer

Step 1 of solving this GMAT Quadratic Algebra Question : Understand Quadratic Equations - Parabola Theory and compute 'c'

y = x² + bx + c is a quadratic equation and the equation represents a parabola.
The curve cuts the y axis at 4.
The x coordinate of the point where it cuts the y axis = 0.
Therefore, (0, 4) is a point on the curve and will satisfy the equation.
Substitute y = 4 and x = 0 in the quadratic equation: 4 = 0² + b(0) + c
Or c = 4.

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Step 2 of solving this GMAT Quadratic Equations Question : Relation between product of roots and 'c'

The product of the roots of a quadratic equation is \\frac{c}{a})
In this question, the product of the roots = \\frac{4}{1}) = 4.

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Step 3 of solving this GMAT Quadratic Equations Question : What do the roots of the quadratic equation represent on the parabola?

The roots of a quadratic equation are the points where the curve (parabola) cuts the x-axis.
The question states that one of the points where the curve cuts the x-axis is -4.
So, -4 is one of roots of the quadratic equation.
Let the second root of the quadratic equation be r₂.
From step 2, we know that the product of the roots of this quadratic equation is 4.
So, -4 * r₂ = 4
or r₂ = -1.

The second root is the second point where the curve cuts the x-axis, which is -1.

The correct choice is (A) and the correct answer is -1.