1

Problem

Solve the equation and check the solutions if it’s satisfied or not: \$| x^2 - 4x - 5| = x + 1\$.

2

Step

The given

\$| x^2 - 4x - 5| = x + 1\$

3

Hint

Set up two cases based on the definition of absolute value: \$x^2 - 4x - 5 = x + 1\$ or \$x^2 - 4x - 5 = - (x + 1)\$.

4

Step

Solve each case \$x^2 - 4x - 5 = x + 1\$and then move all terms to one side:

For \$x^2 - 4x - 5 = x+1\$:

\$x^2 - 4x - 5 - x -1\$

5

Step

Make it simplify

\$x^2 - 5x - 6 = 0\$

6

Step

Factor the quadratic:

\$x^2 - 6x + 1x - 6 = 0\$

\$x(x - 6) + 1( x - 6) = 0\$

\$(x - 6) (x + 1) = 0\$

x = 6 and x = -1

7

Step

Solve each case \$x^2 - 4x - 5 = - (x + 1)\$and then move all terms to one side:

\$x^2 - 4x - 5 = - x - 1\$

\$x^2 - 4x - 5 + x + 1\$

8

Step

Make it simplify

\$x^2 - 3x - 4 = 0\$

9

Step

Factor the quadratic:

\$x^2 - 4x + x - 4 = 0\$

\$x(x - 4) + 1(x - 4) = 0\$

\$(x - 4) ( x + 1) = 0\$

x = 4 and x = -1

10

Step

Check all potential solutions in the original equation:

For x = 6:

\$|6^2 - 4 times 6 - 5\∣ = 6 + 1\$ which simplifies to \$| 7| = 7\$. This is valid

For x = 4:

\$| 4^2 - 4 times 4 - 5| = 4 + 1\$ which simplifies to \$| - 5| = 5\$. This is valid

For x = -1:

\$|1^2 - 4 times 1 - 5| = - 1 + 1\$ which simplifies to∣\$|0\∣ = 0\$. This is valid

11

Solution

Solutions: x = 6, x = - 1, and x = 4 all satisfy the equation \$\∣x^2 - 4x - 5\∣ = x + 1\$.

12

Sumup

Please Summarize Problem, Hint, Clue, Formula and Steps

Choices

13

Choice-A

This is correct: Substituting these values into the equation confirms that the original equation is satisfied

Correct x = 6, x = - 1 and x = 4

14

Choice-B

Substituting these values into the equation doesn’t satisfy the original equation, so Choice B is wrong

Wrong x = -1, x = 4 and x = - 6

15

Choice-C

Substituting these values into the equation doesn’t satisfy the original equation, so there are wrong

Wrong x = -1, x = - 4 and x = 6

16

Choice-D

Substituting these values into the equation doesn’t satisfy the original equation, so option D is wrong

Wrong x = - 6, x = - 4 and x = - 1

17

Answer

Option

A

18

Sumup

Please Summarize Choices