Question

solve the equation. express numbers as integers or simplified fractions.
$4\sqrt{3a + 28} - 4a = 0$
the solution set is .

Explanation:

Step1: Isolate the radical term

$4\sqrt{3a+28} = 4a$

Step2: Simplify both sides by 4

$\sqrt{3a+28} = a$

Step3: Square both sides to eliminate radical

$3a+28 = a^2$

Step4: Rearrange into quadratic form

$a^2 - 3a - 28 = 0$

Step5: Factor the quadratic equation

$(a-7)(a+4) = 0$

Step6: Solve for $a$

$a-7=0 \implies a=7$; $a+4=0 \implies a=-4$

Step7: Check solutions in original equation

For $a=7$: $4\sqrt{3(7)+28}-4(7)=4\sqrt{49}-28=28-28=0$ (valid)
For $a=-4$: $4\sqrt{3(-4)+28}-4(-4)=4\sqrt{16}+16=16+16=32
eq0$ (invalid)

Answer:

$\{7\}$