GMAT Maths | Quadratic Equations #10

This GMAT sample question is a quant problem solving practice question in quadratic equations. It tests your understanding of sum and product of roots of quadratic equations. Interestingly, it also ties in an important number property concept of expressing a number as a product of two of its factors. A 700 level GMAT maths question in number properties and quadratic equations.

Question 10: y = x² + bx + 256 cuts the x axis at (h, 0) and (k, 0). If h and k are integers, what is the least value of b?

1. -32
2. -256
3. -255
4. -257
5. 0

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Video Explanation

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Explanatory Answer

Step 1 of solving this GMAT Quadratic Equations Question: Understand the equation and the points (h, 0) and (k, 0)

The given equation is a quadratic equation. A quadratic equation when plotted on a graph sheet (x - y plane) will result in a parabola.
The roots of the quadratic equation are computed by equating the quadratic expression to 0. i.e., the roots are the values that 'x' take when y = 0
So, the roots of the quadratic equation are the points where the parabola cuts the x-axis.
The question mentions that the curve described by the equation cuts the x-axis at (h, 0) and (k, 0). So, h and k are the roots of the quadratic equation.
For quadratic equations of the form ax² + bx + c = 0, the sum of the roots = \-\frac{b}{a})
The sum of the roots of this equation is \-\frac {b} {1}) = -b.
Note : Higher the value of 'b', i.e., higher the sum of the roots of this quadratic equation, lower the value of b.

For quadratic equations of the form ax² + bx + c = 0, the product of roots = \\frac{c}{a}).
Therefore, the product of the roots of this equation = \\frac{256}{1}) = 256.
i.e., h × k = 256 h and k are both integers.
So, h and k are both integral factors of 256.

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Step 2 of solving this GMAT Quadratic Equations Question: List possible values of h and k and find the least value of ‘b’

This is the step in which number properties concepts kick in. 256 can be expressed as product of two numbers in the following ways:
1 × 256
2 × 128
4 × 64
8 × 32
16 × 16

The sum of the roots is maximum when the roots are 1 and 256 and the maximum sum is 1 + 256 = 257.
∴ The least value possible for b is -257.

Choice D is the correct answer.