If k is a positive real number such that \sqrt{k} + \sqrt{4k} = 15, what is the value of k… — ACT

If $k$ is a positive real number such that $\sqrt{k} + \sqrt{4k} = 15$ , what is the value of $k$ ?

The correct answer is

25

. To find this, we simplify

\sqrt{4k}

2\sqrt{k}

. Substituting this into the equation gives

\sqrt{k} + 2\sqrt{k} = 15

. Combining like terms yields

3\sqrt{k} = 15

. Dividing both sides by

3

gives

\sqrt{k} = 5

. Squaring both sides of the equation results in

k = 25

. Checking the solution:

\sqrt{25} + \sqrt{4(25)} = 5 + 10 = 15

, which is true.

Simplify the radical term

\sqrt{4k}

using the product property of radicals.

\sqrt{4k} = \sqrt{4} \cdot \sqrt{k} = 2\sqrt{k}

This allows us to express both terms in the equation using the same radical base,

\sqrt{k}

Substitute the simplified term back into the equation and combine like terms.

\sqrt{k} + 2\sqrt{k} = 15 \implies 3\sqrt{k} = 15

Combining like radical terms simplifies the equation to a single radical term.

Isolate the radical by dividing both sides of the equation by

3

\sqrt{k} = 5

Isolating the radical term is necessary before squaring both sides to solve for the variable.

Square both sides of the equation to solve for

k

k = 25

Squaring is the inverse operation of taking the square root, which isolates

k

Solving radical equations by simplifying and combining like radical terms

Soru