For all x \ge 0, the expression (3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5) is equivalent… — SAT

For all $x \ge 0$ , the expression $(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)$ is equivalent to $ax - b\sqrt{x} - 10$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Cevap: 17

Cevap

Expanding the expression

(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)

yields

6x - 15x^{\frac{1}{2}} + 4x^{\frac{1}{2}} - 10

. Combining like terms gives

6x - 11\sqrt{x} - 10

. Comparing this result to the given equivalent form

ax - b\sqrt{x} - 10

shows that

a = 6

and

b = 11

. Therefore, the value of

a + b

6 + 11 = 17

Adım Adım Çözüm

Multiply the binomials

(3x^{\frac{1}{2}} + 2)(2x^{\frac{1}{2}} - 5)

using FOIL.

(3x^{\frac{1}{2}})(2x^{\frac{1}{2}}) - 5(3x^{\frac{1}{2}}) + 2(2x^{\frac{1}{2}}) - 10

Expanding the product allows us to write it in the standard simplified form.

Simplify the products and combine the like terms.

6x - 15x^{\frac{1}{2}} + 4x^{\frac{1}{2}} - 10 = 6x - 11x^{\frac{1}{2}} - 10

This simplifies the expression so we can compare it directly to the target form.

Rewrite

x^{\frac{1}{2}}

\sqrt{x}

and compare the expression to

ax - b\sqrt{x} - 10

6x - 11\sqrt{x} - 10 = ax - b\sqrt{x} - 10

, which gives

a = 6

and

b = 11

Matching corresponding coefficients allows us to determine the values of the constants

a

and

b

Calculate the sum of

a

and

b

a + b = 6 + 11 = 17

The question requires finding the value of

a + b

Anahtar Kavram

To determine equivalence between algebraic expressions, expand the terms using the distributive property, simplify, and equate the corresponding coefficients of the like terms.

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