If 16^{x-1} - 10 \cdot 4^{x-2} + 1 = 0, what is the sum of all values of x that satisfy th… — SAT

If $16^{x-1} - 10 \cdot 4^{x-2} + 1 = 0$ , what is the sum of all values of $x$ that satisfy the equation?

Answer: 2

The sum of all values of

x

that satisfy the equation is 2.

By writing

16^{x-1}

\frac{(4^x)^2}{16}

and

10 \cdot 4^{x-2}

\frac{10 \cdot 4^x}{16}

, multiplying the entire equation by 16 yields

(4^x)^2 - 10(4^x) + 16 = 0

. Factoring this quadratic equation gives

(4^x - 2)(4^x - 8) = 0

, which yields

4^x = 2

and

4^x = 8

. Solving these exponential equations results in

x = 0.5

and

x = 1.5

. Adding these values together gives

0.5 + 1.5 = 2

Express each exponential term in the equation using a base of 4.

\frac{4^{2x}}{16} - \frac{10 \cdot 4^x}{16} + 1 = 0

Since

16 = 4^2

, we can write

16^{x-1} = (4^2)^{x-1} = 4^{2x-2} = \frac{4^{2x}}{16}

. Similarly,

4^{x-2} = \frac{4^x}{16}

Multiply every term in the equation by 16 to eliminate the fractions.

4^{2x} - 10 \cdot 4^x + 16 = 0

This simplifies the expression into a standard polynomial form that is quadratic in terms of

4^x

Use the substitution

u = 4^x

(where

u > 0

) to transform the equation into a quadratic equation in terms of

u

, and solve it.

u^2 - 10u + 16 = 0 \implies (u - 2)(u - 8) = 0 \implies u = 2

u = 8

Substituting

u

allows us to factor the quadratic expression using standard factoring techniques.

Substitute

4^x

back for

u

and solve for

x

using base 2 conversion.

For

u = 2

4^x = 2 \implies 2^{2x} = 2^1 \implies x = 0.5

. For

u = 8

4^x = 8 \implies 2^{2x} = 2^3 \implies x = 1.5

We must convert the solutions for the intermediate variable

u

back to the original variable

x

Calculate the sum of all found values of

x

0.5 + 1.5 = 2

The question asks for the sum of all real values of

x

that satisfy the equation.

Solving exponential equations of quadratic form by using variable substitution and exponent laws.

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