Question

Given quadratic equation is \( x^{2} - |x| - 30 = 0 \). Then which of the following statements is/are incorrect?

• A. \( x - 6 = 0 \)
• B. \( x + 6 = 0 \)
• C. \( x + 5 = 0 \)
• D. Both (c) and (d)

Hint:

Substitute \( u = |x| \) to turn the equation into a simple quadratic in \( u \), then check which linear statements actually match the roots.

Answer 1

The Correct Option is D

Step 1: Substitute for the absolute value.
Let \( u = |x| \), so \( u \geq 0 \), and the equation becomes \( u^{2} - u - 30 = 0 \).

Step 2: Factor and solve for \( u \).
\( (u-6)(u+5) = 0 \), so \( u = 6 \) or \( u = -5 \).
Reject \( u = -5 \) since \( |x| \) can never be negative.

Step 3: Find the roots of \( x \).
\( |x| = 6 \) gives \( x = 6 \) or \( x = -6 \).

Step 4: Check every printed statement against these roots.
(a) \( x - 6 = 0 \) gives \( x = 6 \), a genuine root, so this statement is correct.
(b) \( x + 6 = 0 \) gives \( x = -6 \), also a genuine root, so this statement is correct.
(c) \( x + 5 = 0 \) gives \( x = -5 \), which is not a root, so this statement is incorrect.
(d) \( x + 7 = 0 \) gives \( x = -7 \), also not a root, so this statement is incorrect too.

Final Answer:
Both (c) and (d) are the incorrect statements. \[ \boxed{\text{Both (c) and (d)}} \]