Q. \(x^2+4=0\)

Q: How do complex numbers solve x^2=-4?

Take square roots: \sqrt{-4}=2i. Thus x=\pm 2i.

Q: What is the discriminant of x^2+4=0?

Write a=1, b=0, c=4. Then \Delta=b^2-4ac=0-16=-16.

Q: Why are there no real solutions?

Real solutions require x^2=-4, but x^2\ge 0 for real x. Hence no real x.

Q: What is the graph interpretation of x^2+4=0?

y=x^2+4 is shifted up by 4, so y\ge 4. It never hits y=0. Complex intercepts are \pm 2i.

Q: Can we factor x^2+4?

No real factorization. Over complex numbers: x^2+4=(x-2i)(x+2i).

Answer

We solve the quadratic equation \(x^2+4=0\).

\[
x^2=-4
\]
Since \(-4=4(-1)\), we get \(x^2=4(-1)\), so \(x=\pm 2i\).

Final result: \(x=2i\) or \(x=-2i\).

Detailed Explanation

We want to solve the equation

\[
x^2+4=0
\]

Step 1: Isolate the \(x^2\) term by subtracting \(4\) from both sides.

\[
x^2+4-4=0-4
\]

\[
x^2=-4
\]

Step 2: Take a square root of both sides. Remember that square roots of a negative number involve the imaginary unit \(i\), where \(i^2=-1\).

\[
x=\pm\sqrt{-4}
\]

Step 3: Rewrite \(-4\) in terms of \(-1\).

\[
\sqrt{-4}=\sqrt{(-1)\cdot 4}=\sqrt{-1}\,\sqrt{4}
\]

\[
\sqrt{-1}=i,\qquad \sqrt{4}=2
\]

\[
\sqrt{-4}=2i
\]

Step 4: Include the \(\pm\) from taking the square root.

\[
x=\pm 2i
\]

Final answer:

\[
x=2i \quad \text{or} \quad x=-2i
\]

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Algebra FAQ

Solve \(x^2+4=0\).

\(x^2=-4\), so \(x=\pm 2i\).

How do complex numbers solve \(x^2=-4\)?

Take square roots: \(\sqrt{-4}=2i\). Thus \(x=\pm 2i\).

What is the discriminant of \(x^2+4=0\)?

Write \(a=1\), \(b=0\), \(c=4\). Then \(\Delta=b^2-4ac=0-16=-16\).

Why are there no real solutions?

Real solutions require \(x^2=-4\), but \(x^2\ge 0\) for real \(x\). Hence no real \(x\).

How to use the quadratic formula on \(x^2+4=0\)?

\(x=\dfrac{-b\pm\sqrt{\Delta}}{2a}=\dfrac{0\pm\sqrt{-16}}{2}=\pm 2i\).

What is the graph interpretation of \(x^2+4=0\)?

\(y=x^2+4\) is shifted up by 4, so \(y\ge 4\). It never hits \(y=0\). Complex intercepts are \(\pm 2i\).

Can we factor \(x^2+4\)?

No real factorization. Over complex numbers: \(x^2+4=(x-2i)(x+2i)\).

Solve \(x^2+4=0\) step by step.
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