Let f(x) = x^2 + bx + c, where b and c are integers. The equation f(x) = 0 has two distinc… — GRE General Test

Let $f(x) = x^2 + bx + c$ , where $b$ and $c$ are integers. The equation $f(x) = 0$ has two distinct real roots, $\alpha$ and $\beta$ . If $\alpha + \beta = \alpha\beta$ and $c > 0$ , which of the following statements must be true? Indicate all such statements.

The constant $c$ is strictly greater than 4.Answer
Both roots $\alpha$ and $\beta$ are positive.Answer
The vertex of the parabola $y = f(x)$ lies in Quadrant IV of the xy-plane.Answer
D
The coefficient $b$ is positive.
E
The difference between the two roots, $|\alpha - \beta|$ , can equal 2.

Answer

The statements asserting that the constant

c

is strictly greater than 4, both roots are positive, and the vertex lies in Quadrant IV must all be true.

By Vieta's formulas,

\alpha + \beta = -b

and

\alpha\beta = c

. Equating sum and product gives

b = -c

. The discriminant

\Delta = c^2 - 4c > 0

requires

c > 4

because

c > 0

. Since sum and product of the roots equal

c > 0

, both roots are positive. The vertex coordinates

(c/2, -c(c-4)/4)

have a positive x-value and a negative y-value, placing the vertex in Quadrant IV.

Step-by-Step Solution

Apply Vieta's formulas and establish the relationship between coefficients

\alpha + \beta = -b

and

\alpha\beta = c

. Setting them equal gives

-b = c \implies b = -c

The stem specifies that the sum of the roots equals their product.

Analyze the discriminant for distinct real roots

\Delta = b^2 - 4c = (-c)^2 - 4c = c(c - 4) > 0

. Since

c > 0

c - 4 > 0 \implies c > 4

Two distinct real roots require a strictly positive discriminant.

Determine the signs of the roots

\alpha + \beta = c > 0

and

\alpha\beta = c > 0

, implying

\alpha > 0

and

\beta > 0

If the sum and product of two real numbers are both positive, both numbers must be positive.

Find the location of the parabola's vertex

x_v = \frac{c}{2} > 0

and

y_v = -\frac{c(c-4)}{4} < 0

, placing the vertex in Quadrant IV.

A point with a positive x-coordinate and negative y-coordinate resides in the fourth quadrant.

Evaluate the remaining options regarding

b

and

|\alpha - \beta|

b = -c < -4

(negative), and

|\alpha - \beta| = 2 \implies c = 2 + 2\sqrt{2}

, which is not an integer.

These evaluations disprove the statements that

b

is positive and that

|\alpha - \beta|

can equal 2.

Key Concept

Quadratic Root Properties and Vieta's Formulas

Question

Answer

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