This document discusses solving the quadratic equation x2 – 7 |x| – 18 = 0. It splits the problem into two cases: when x is greater than 0 and when x is less than 0. For each case, it is shown that the only real solution is x = 9. Therefore, the number of real solutions to the given equation is 2, as indicated by choice (a).

1. Quadratic Equations

2. Quadratic Equations
What is the number of real solutions of the equation x2 – 7 |x| – 18 =
0?
(a) 2 (b) 4
(c) 3 (d) 1

3. Quadratic Equations
What is the number of real solutions of the equation x2 – 7 |x| – 18 =
0?
Let us split this into two cases. Case 1, when x is greater than 0 and
Case 2, when x is lesser than 0.
Case 1: x > 0. Now, |x| = x
x2 – 7x – 18 = 0
(x – 9) (x + 2) = 0
x is either –2 or +9.

4. Quadratic Equations
What is the number of real solutions of the equation x2 – 7 |x| – 18 =
0?
However, in accordance with the initial assumption that x > 0, x can
only be + 9 (cannot be –2).
Case 2:
x < 0. Now, |x| = –x
x2 + 7x – 18 = 0
(x + 9) (x – 2) = 0
x is either –9 or +2.

5. Quadratic Equations
What is the number of real solutions of the equation x2 – 7 |x| – 18 =
0?
However, in accordance with the initial assumption that x < 0, x can
only be –9 (cannot be +2).
Hence, this equation has two roots: –9 and +9.
Alternatively, we can treat this as a quadratic in |x|, the equation
can be written as |x|2 – 7 |x| – 18 = 0.
Or, (|x| – 9) (|x| + 2) = 0
|x| = 9 or –2. |x| cannot be –2.
|x| = 9, x = 9 or –9.
Answer choice (a)

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