If x is a real number such that (x^2 - 5x + 5)^{x^2 - 9x + 20} = 1, what is the sum of all… — GMAT

If $x$ is a real number such that $(x^2 - 5x + 5)^{x^2 - 9x + 20} = 1$ , what is the sum of all distinct real values of $x$ that satisfy the equation?

A
10
B
12
C
14
15Answer
E
9

Answer

The sum of all distinct real values of

x

that satisfy the equation is 15.

The equation

(x^2 - 5x + 5)^{x^2 - 9x + 20} = 1

holds under three distinct conditions: when the exponent is 0 and the base is non-zero (

x = 4, 5

), when the base is 1 (

x = 1, 4

), and when the base is

-1

provided the exponent is an even integer (

x = 2, 3

, yielding even exponents 6 and 2 respectively). The set of distinct solutions is

\{1, 2, 3, 4, 5\}

, and their sum is 15.

Step-by-Step Solution

Analyze the general algebraic conditions under which an expression of the form

u^v = 1

holds for real numbers

u

and

v

The equation

u^v = 1

is satisfied in three distinct cases: (1)

v = 0

and

u \neq 0

, (2)

u = 1

, or (3)

u = -1

with

v

being an even integer.

Any non-zero real number raised to the power of 0 equals 1, 1 raised to any power equals 1, and

-1

raised to an even integer power equals 1.

Evaluate Case 1 where the exponent

x^2 - 9x + 20 = 0

Factoring gives

(x - 4)(x - 5) = 0

, yielding

x = 4

and

x = 5

. Testing the base

x^2 - 5x + 5

x = 4

gives

1 \neq 0

, and at

x = 5

gives

5 \neq 0

. Thus

x = 4

and

x = 5

are valid solutions.

Exponent equal to zero produces

u^0 = 1

as long as the base

u \neq 0

Evaluate Case 2 where the base

x^2 - 5x + 5 = 1

Subtracting 1 gives

x^2 - 5x + 4 = 0

, which factors as

(x - 1)(x - 4) = 0

, yielding

x = 1

and

x = 4

. Both are valid.

Base equal to 1 raised to any real exponent equals 1.

Evaluate Case 3 where the base

x^2 - 5x + 5 = -1

and check if the exponent is an even integer.

Adding 1 gives

x^2 - 5x + 6 = 0

, which factors as

(x - 2)(x - 3) = 0

, yielding

x = 2

and

x = 3

. Testing

x = 2

in the exponent gives

2^2 - 9(2) + 20 = 6

(even). Testing

x = 3

gives

3^2 - 9(3) + 20 = 2

(even). Thus

x = 2

and

x = 3

are also valid solutions.

Base

-1

raised to an even integer power equals 1.

Combine all unique real values of

x

and compute their sum.

The set of distinct real solutions is

\{1, 2, 3, 4, 5\}

. Their sum is

1 + 2 + 3 + 4 + 5 = 15

Summing all identified distinct valid roots.

Key Concept

Solving polynomial exponential equations of the form

f(x)^{g(x)} = 1

by systematically testing base-exponent cases.

Estimated Time:2m 0s

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Step-by-Step Solution

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