What is the sum of all real solutions to the equation \sqrt{2x^2 - 5x + 3} = 1 - x? — GMAT

What is the sum of all real solutions to the equation $\sqrt{2x^2 - 5x + 3} = 1 - x$ ?

Squaring both sides of

\sqrt{2x^2 - 5x + 3} = 1 - x

yields

2x^2 - 5x + 3 = (1 - x)^2 = x^2 - 2x + 1

. Moving all terms to one side yields the quadratic equation

x^2 - 3x + 2 = 0

, which factors as

(x - 1)(x - 2) = 0

. The algebraic solutions are

x = 1

and

x = 2

. Testing

x = 1

in the original equation gives

\sqrt{0} = 0

, which is valid. Testing

x = 2

gives

\sqrt{1} = -1

, which is invalid because the principal square root cannot be negative. Thus,

x = 1

is the sole valid real solution, and its sum is 1.

Square both sides of the equation to eliminate the radical

2x^2 - 5x + 3 = (1 - x)^2 = 1 - 2x + x^2

Squaring both sides converts the radical equation into a standard algebraic polynomial equation.

Rearrange into standard quadratic form

ax^2 + bx + c = 0

x^2 - 3x + 2 = 0

Subtracting

(x^2 - 2x + 1)

from both sides collects all terms on one side.

Factor the quadratic expression to find candidate roots

(x - 1)(x - 2) = 0 \implies x = 1 \text{ or } x = 2

Factoring determines the values of

x

that satisfy the squared equation.

Substitute candidate roots back into the original equation to check for extraneous solutions

For

x = 1

\sqrt{2(1)^2 - 5(1) + 3} = \sqrt{0} = 0

, and

1 - 1 = 0

(Valid).
For

x = 2

\sqrt{2(2)^2 - 5(2) + 3} = \sqrt{1} = 1

, but

1 - 2 = -1

(Invalid, since

1 \neq -1

Squaring an equation can introduce extraneous roots where the principal square root would equal a negative quantity.

Sum the valid real solutions

The only valid solution is

x = 1

, so the sum is 1.

Only valid roots that satisfy the original equation may be summed.

Quadratic Factoring and Extraneous Solutions in Radical Equations

Estimated Time:1m 30s

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