Questions: Solve the following equation and choose the best possible answer below. 3 x^2+48=0 Select one: a. x= ±-2 i b. x= ± 4 i c. x= ± 10 i d. x= ± 5 i

Transcript text: Solve the following equation and choose the best possible answer below. \[ 3 x^{2}+48=0 \] Select one: a. $x= \pm-2 i$ b. $x= \pm 4 i$ c. $x= \pm 10 i$ d. $x= \pm 5 i$

Solution

Solution Steps

To solve the quadratic equation $3x^2 + 48 = 0$, we first isolate $x^2$ by subtracting 48 from both sides and then dividing by 3. This gives us $x^2 = -16$. Since the square of a real number cannot be negative, we introduce the imaginary unit $i$, where $i^2 = -1$. Thus, $x^2 = -16$ becomes $x^2 = 16i^2$, and taking the square root of both sides gives $x = \pm 4i$.

Step 1: Rewrite the Equation

The given equation is: \[ 3x^2 + 48 = 0 \]

Step 2: Isolate $x^2$

Subtract 48 from both sides: \[ 3x^2 = -48 \]

Divide both sides by 3: \[ x^2 = -16 \]

Step 3: Solve for $x$

Since $x^2 = -16$, we introduce the imaginary unit $i$, where $i^2 = -1$. Thus, we have: \[ x^2 = 16i^2 \]

Taking the square root of both sides: \[ x = \pm 4i \]