Question

The roots of the quadratic equation $(x - 1)^2 = 16$ are :

• A. $5, 3$
• B. $4, -4$
• C. $5, -3$
• D. $-5, 3$

Hint:

Always remember that taking the square root of a positive real number yields both positive and negative roots.
A common mistake is to write $x - 1 = 4$ only, which leads to losing one of the valid roots of the quadratic equation.
You can also quickly verify your answer by substituting the options back into the original equation:
For $x = 5 \implies (5 - 1)^2 = 4^2 = 16$ (Correct)
For $x = -3 \implies (-3 - 1)^2 = (-4)^2 = 16$ (Correct)

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question belongs to the topic of "Quadratic Equations".
A quadratic equation is a second-degree polynomial equation in a single variable.
The general form of a quadratic equation is given by $ax^2 + bx + c = 0$, where $a \neq 0$.
Here, we are given a quadratic equation in a simplified perfect square form, and we need to determine its two roots.
The roots of an equation are the values of the variable $x$ that satisfy the given algebraic expression.

Step 2: Key Formula or Approach:
When we have a quadratic equation already expressed in the form $(x - a)^2 = b$, where $b \ge 0$, we can solve for $x$ directly by taking the square root on both sides.
The mathematical operation of taking the square root yields both a positive and a negative value:
\[ x - a = \pm\sqrt{b} \]
This allows us to split the equation into two separate first-degree linear equations:
\[ x - a = \sqrt{b} \quad \text{and} \quad x - a = -\sqrt{b} \]
We then solve each linear equation individually to obtain the two distinct roots.

Step 3: Detailed Explanation:

• Write down the given quadratic equation:
\[ (x - 1)^2 = 16 \]

• Take the square root on both sides of the equation to eliminate the exponent:
\[ \sqrt{(x - 1)^2} = \pm\sqrt{16} \]

• Since the square root of 16 is 4, this simplifies to:
\[ x - 1 = \pm 4 \]

• Now, split this expression into two separate cases based on the plus-minus sign:

• Case 1 (Using the positive sign):
\[ x - 1 = 4 \]
Add 1 to both sides of the equation:
\[ x = 4 + 1 \implies x = 5 \]

• Case 2 (Using the negative sign):
\[ x - 1 = -4 \]
Add 1 to both sides of the equation:
\[ x = -4 + 1 \implies x = -3 \]

• Therefore, the two roots of the quadratic equation are $x = 5$ and $x = -3$.


Step 4: Final Answer:
By comparing our derived roots with the given options, we find that the roots are $5$ and $-3$, which corresponds to Option (C).