Question

Given quadratic equation is x² - |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 solve the problem, we begin by analyzing the given quadratic equation:

\(x^2 - |x| - 30 = 0\) 

This is not a standard quadratic equation due to the presence of the absolute value term. To proceed, we need to examine it by considering two cases based on the value of \(x\):

1. Case 1: \(x \geq 0\)

When \(x\) is non-negative, \(|x| = x\). Thus, the equation becomes:

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

We solve for \(x\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -1\), and \(c = -30\).

Calculating the discriminant:

\(b^2 - 4ac = (-1)^2 - 4 \times 1 \times (-30) = 1 + 120 = 121\)

The solutions are:

\(x = \frac{1 \pm \sqrt{121}}{2} = \frac{1 \pm 11}{2}\)

This gives us:

• \(x_1 = \frac{1 + 11}{2} = 6\)
• \(x_2 = \frac{1 - 11}{2} = -5\)

1. Case 2: \(x < 0\)

When \(x\) is negative, \(|x| = -x\). The equation becomes:

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

Using the quadratic formula again with \(a = 1\), \(b = 1\), and \(c = -30\), we find:

Calculating the discriminant:

\(b^2 - 4 \cdot 1 \cdot (-30) = 1 + 120 = 121\)

The solutions are:

\(x = \frac{-1 \pm \sqrt{121}}{2} = \frac{-1 \pm 11}{2}\)

This leads to:

• \(x_1 = \frac{-1 + 11}{2} = 5\)
• \(x_2 = \frac{-1 - 11}{2} = -6\)

Based on the above analysis, the roots of the equation are \(6, -5,\) and \(-6\). We will now check the incorrect statements among the options provided:

• \(x - 6 = 0\) is a correct statement as \(x = 6\) is a solution.
• \(x + 6 = 0\) is a correct statement as \(x = -6\) is a solution.
• \(x + 5 = 0\) is incorrect because \(x = -5\) is a solution.
• \(x + 7 = 0\) is incorrect because neither \(x = -7\) nor \(x = 7\) are solutions.

Thus, the options that are incorrect are:

Both (c) and (d).