What is the sum of all real solutions to the equation \sqrt{3x + 7} - \frac{10}{\sqrt{3x +… — SAT

What is the sum of all real solutions to the equation $\sqrt{3x + 7} - \frac{10}{\sqrt{3x + 7}} = 3$ ?

Cevap: 6

Cevap

The only real solution is 6, so the sum of all real solutions is 6.

The correct answer is 6. By substituting

u = \sqrt{3x + 7}

, the equation becomes

u - 10/u = 3

, which simplifies to the quadratic

u^2 - 3u - 10 = 0

. Factoring gives

(u - 5)(u + 2) = 0

. Since the principal square root must be non-negative,

u

must be positive, so we reject

u = -2

and keep

u = 5

. Solving

\sqrt{3x + 7} = 5

by squaring both sides yields

3x + 7 = 25

, which simplifies to

x = 6

. The extraneous solution

x = -1

(which comes from

u = -2

) must be discarded because substituting it back into the original equation results in

-3

instead of

3

. Thus, the only real solution is 6.

Adım Adım Çözüm

Substitute

u = \sqrt{3x + 7}

into the equation.

u - \frac{10}{u} = 3

, where

u > 0

This substitution simplifies the radical equation into a rational equation of a single variable.

Multiply the entire equation by

u

and solve the resulting quadratic equation.

u^2 - 3u - 10 = 0 \implies (u - 5)(u + 2) = 0

, which yields

u = 5

u = -2

Clearing the denominator transforms the rational expression into a standard quadratic equation.

Apply the constraint that

u = \sqrt{3x + 7}

must be non-negative.

u = 5

is the only valid root;

u = -2

has no real solution and must be discarded.

The output of a principal square root is non-negative by definition.

Substitute back

\sqrt{3x + 7} = 5

and solve for

x

3x + 7 = 25 \implies 3x = 18 \implies x = 6

Squaring both sides eliminates the radical and allows us to isolate the variable

x

Verify if solving

\sqrt{3x+7} = -2

leads to an extraneous solution.

Squaring both sides of

\sqrt{3x+7} = -2

gives

3x+7 = 4 \implies x = -1

. Substituting

x = -1

into the original equation yields

\sqrt{4} - \frac{10}{\sqrt{4}} = 2 - 5 = -3 \neq 3

. Therefore,

x = -1

is extraneous.

Verifying solutions is necessary to identify and discard values that do not satisfy the original equation.

Anahtar Kavram

Solving equations using substitution and identifying extraneous solutions

Tahmini Süre:2m 30s

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