The quadratic equation (x - a)(x - b) = c, where a, b, and c are real constants with a < b… — GRE General Test

The quadratic equation $(x - a)(x - b) = c$ , where $a$ , $b$ , and $c$ are real constants with $a < b$ and $c > 0$ , has two real roots $x_1$ and $x_2$ such that $x_1 < x_2$ . Which of the following statements must be true? Indicate all such statements.

$x_1 < a$ Cevap
$x_2 > b$ Cevap
The distance between the roots, $x_2 - x_1$ , is strictly greater than $b - a$ Cevap
D
$x_1 + x_2 > a + b$
E
The product of the roots, $x_1 x_2$ , must be negative

Cevap

The correct statements are that the smaller root is less than

a

(

x_1 < a

), the larger root is greater than

b

(

x_2 > b

), and the distance between the roots is greater than

b - a

(

x_2 - x_1 > b - a

Evaluating

f(x) = (x-a)(x-b) - c

x = a

and

x = b

yields negative values (

-c

). Because the parabola opens upward, the graph must cross the x-axis to the left of

a

and to the right of

b

. This establishes

x_1 < a

and

x_2 > b

. Combining these inequalities shows that the distance between the roots

x_2 - x_1

must exceed

b - a

Adım Adım Çözüm

Formulate the quadratic function and analyze its values at key points

Let

f(x) = (x - a)(x - b) - c = 0

. Evaluating at

x = a

and

x = b

gives

f(a) = (0)(a-b) - c = -c < 0

and

f(b) = (b-a)(0) - c = -c < 0

Evaluating at

a

and

b

reveals the sign of the function inside the interval

[a, b]

Use the sign of the leading coefficient and intermediate value properties to locate the roots

The coefficient of

x^2

1 > 0

, so the parabola opens upward. Since

f(a) < 0

and

f(b) < 0

, and

f(x) \to +\infty

x \to \pm\infty

, the two real roots

x_1

and

x_2

must satisfy

x_1 < a < b < x_2

A continuous upward-opening parabola must cross the x-axis outside any interval where its values are negative.

Analyze the distance between the roots

Subtracting

x_1 < a

from

x_2 > b

gives

x_2 - x_1 > b - a

Since

a

and

b

fall strictly between

x_1

and

x_2

, the distance between

x_1

and

x_2

exceeds the distance between

a

and

b

Check the remaining statements using Vieta's formulas

Expanding

f(x) = x^2 - (a+b)x + (ab - c) = 0

yields sum of roots

x_1 + x_2 = a + b

and product of roots

x_1 x_2 = ab - c

. Thus

x_1 + x_2 > a+b

is false, and

x_1 x_2 < 0

is not necessarily true.

Vieta's formulas give exact values for the sum and product of roots in terms of coefficients.

Anahtar Kavram

Geometric interpretation of quadratic functions and root locations relative to evaluated points

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