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. x + 7 = 0
• E. Both (c) and (d)

Answer 1

Answer: (E)

To find which statements are incorrect regarding the given quadratic equation \(x^2 - |x| - 30 = 0\), we need to analyze the possible roots of this equation.

The equation given is \(x^2 - |x| - 30 = 0\).

We consider two cases because of the absolute value function:

1. Case 1: When \(x \geq 0\), |x| = x.
2. Case 2: When \(x < 0\), |x| = -x.

For Case 1 (\(x \geq 0\)):

The equation becomes:

\(x^2 - x - 30 = 0\)

Factorizing the equation:

\(x^2 - x - 30 = (x - 6)(x + 5) = 0\)

This gives the roots \(x = 6\) and \(x = -5\).

For Case 2 (\(x < 0\)):

The equation becomes:

\(x^2 + x - 30 = 0\)

Factorizing the equation:

\(x^2 + x - 30 = (x - 5)(x + 6) = 0\)

This gives the roots \(x = 5\) and \(x = -6\), but only \(x = -6\) is valid for \(x < 0\).

We conclude that the valid roots of the original equation \(x^2 - |x| - 30 = 0\) are \(x = 6\), \(x = -5\), and \(x = -6\).

The options given are:

1. \(x - 6 = 0\) → Equivalent to \(x = 6\) (a correct root)
2. \(x + 6 = 0\) → Equivalent to \(x = -6\) (a correct root)
3. \(x + 5 = 0\) → Equivalent to \(x = -5\) (a correct root)
4. \(x + 7 = 0\) → Equivalent to \(x = -7\) (incorrect root)

Thus, the incorrect statements are:

• \(x + 7 = 0\)
• Option (c) is incorrect as it does not match with any root.
• Therefore, both Option (c) and (d) are incorrect as per this analysis.

So, the correct answer is: Both (c) and (d).