Question

Which equation has no real solutions?

A. –2x2 = –168
B. 4x2 + 76 = 32
C. 25x2 = 1
D. 43 − x2 = 12

Answer 1

To determine which equation has no real solutions, we can analyze each option step by step.

**A.** \(-2x^2 = -168\)

Dividing both sides by -2:
\[
x^2 = 84
\]
Taking the square root:
\[
x = \pm \sqrt{84}
\]
So, this has real solutions.

---

**B.** \(4x^2 + 76 = 32\)

Subtracting 76 from both sides:
\[
4x^2 = 32 - 76
\]
\[
4x^2 = -44
\]
Dividing by 4:
\[
x^2 = -11
\]
This equation has no real solutions because a square cannot be negative.

---

**C.** \(25x^2 = 1\)

Dividing both sides by 25:
\[
x^2 = \frac{1}{25}
\]
Taking the square root:
\[
x = \pm \frac{1}{5}
\]
So, this has real solutions.

---

**D.** \(43 - x^2 = 12\)

Subtracting 43 from both sides:
\[
-x^2 = 12 - 43
\]
\[
-x^2 = -31
\]
Multiplying by -1:
\[
x^2 = 31
\]
Taking the square root:
\[
x = \pm \sqrt{31}
\]
So, this has real solutions.

---

The only equation with no real solutions is **B. \(4x^2 + 76 = 32\)**.