Solve the radical equation, and check all proposed solutions.

Question

Solve the radical equation, and check all proposed solutions. $$ \sqrt{21 x-45}=x+3 $$

UpStudy AI · Accepted Answer

We start with the equation

$$
\sqrt{21x-45} = x + 3.
$$

**Step 1. Determine the domain**

1. The expression inside the square root must be nonnegative:

$$
   21x - 45 \ge 0 \quad \Rightarrow \quad x \ge \frac{45}{21} = \frac{15}{7}.
   $$

2. The square root outputs a nonnegative value, so the right-hand side must also be nonnegative:

$$
   x + 3 \ge 0 \quad \Rightarrow \quad x \ge -3.
   $$

Thus, the overall domain is

$$
x \ge \frac{15}{7}.
$$

**Step 2. Square both sides to remove the square root**

Squaring both sides gives

$$
21x - 45 = (x+3)^2.
$$

Expanding the right-hand side:

$$
21x - 45 = x^2 + 6x + 9.
$$

**Step 3. Rearrange the equation to standard quadratic form**

Bring all terms to one side:

$$
0 = x^2 + 6x + 9 - 21x + 45.
$$

Combine like terms:

$$
0 = x^2 - 15x + 54.
$$

**Step 4. Solve the quadratic equation**

The quadratic equation is

$$
x^2 - 15x + 54 = 0.
$$

We can factor the quadratic:

$$
x^2 - 15x + 54 = (x-6)(x-9)=0.
$$

Thus, the potential solutions are

$$
x = 6 \quad \text{and} \quad x = 9.
$$

**Step 5. Check solutions in the original equation**

1. For $$ x = 6 $$:

$$
   \sqrt{21(6)-45} = \sqrt{126-45} = \sqrt{81} = 9,
   $$
   
   and
   
   $$
   6 + 3 = 9.
   $$
   
   The left-hand side equals the right-hand side.

2. For $$ x = 9 $$:

$$
   \sqrt{21(9)-45} = \sqrt{189-45} = \sqrt{144} = 12,
   $$
   
   and
   
   $$
   9 + 3 = 12.
   $$
   
   Again, the left-hand side equals the right-hand side.

Both solutions satisfy the original equation as well as the domain conditions.

**Answer**

$$
x = 6 \quad \text{or} \quad x = 9.
$$
$$ x = 6 $$ or $$ x = 9 $$.

Solve the radical equation, and check all proposed solutions.

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