Solve the radical equation, and check all proposed solutions,

Question

Solve the radical equation, and check all proposed solutions, $$ \sqrt{25 x-54}=x+4 $$

UpStudy AI · Accepted Answer

**Step 1. Determine the Domain**

For the radical expression $$ \sqrt{25x - 54} $$ to be defined, the radicand must be nonnegative:
$$
25x - 54 \geq 0 \quad \Rightarrow \quad x \geq \frac{54}{25}.
$$
Also, since the square root yields a nonnegative number, the right-hand side $$ x+4 $$ must be nonnegative:
$$
x+4 \geq 0 \quad \Rightarrow \quad x \geq -4.
$$
Thus, the effective domain is $$ x \geq \frac{54}{25} $$ (since $$\frac{54}{25} \approx 2.16 > -4$$).

**Step 2. Square Both Sides**

Given the equation
$$
\sqrt{25x - 54} = x+4,
$$
we square both sides to remove the square root:
$$
25x - 54 = (x+4)^2.
$$

**Step 3. Expand and Rearrange**

First, expand the right-hand side:
$$
(x+4)^2 = x^2 + 8x + 16.
$$
Now substitute back:
$$
25x - 54 = x^2 + 8x + 16.
$$
Bring all terms to one side:
$$
0 = x^2 + 8x + 16 - 25x + 54,
$$
which simplifies to:
$$
x^2 - 17x + 70 = 0.
$$

**Step 4. Solve the Quadratic Equation**

Factor the quadratic:
$$
x^2 - 17x + 70 = (x-7)(x-10) = 0.
$$
Thus, the potential solutions are:
$$
x=7 \quad \text{or} \quad x=10.
$$

**Step 5. Verify Both Solutions**

1. **For $$ x=7 $$:**
   - Check the radicand:
     $$
     25(7) - 54 = 175 - 54 = 121 \geq 0.
     $$
   - Substitute into the original equation:
     $$
     \sqrt{121} = 11 \quad \text{and} \quad 7+4 = 11.
     $$
   - Both sides match.

2. **For $$ x=10 $$:**
   - Check the radicand:
     $$
     25(10) - 54 = 250 - 54 = 196 \geq 0.
     $$
   - Substitute into the original equation:
     $$
     \sqrt{196} = 14 \quad \text{and} \quad 10+4 = 14.
     $$
   - Both sides match.

**Conclusion**

Both solutions satisfy the original equation and the domain restrictions. Hence, the solutions are:
$$
\boxed{x=7,\quad x=10.}
$$
The solutions are $$ x = 7 $$ and $$ x = 10 $$.

Solve the radical equation, and check all proposed solutions,

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