Q. \(x^2 + 9 = 0\)

Q: How do you solve x^2+9=0?

Rearrange: x^2=-9. Take square roots: x=\pm \sqrt{-9}=\pm 3i.

Q: Why are the solutions imaginary for x^2+9=0?

Because x^2=-9, and a real square cannot be negative. So x must be complex, namely x=\pm 3i.

Q: What does \sqrt{-9} equal?

Use \sqrt{-a}=i\sqrt{a} . So \sqrt{-9}=i\sqrt{9}=3i.

Q: How do you check x=3i satisfies the equation?

Substitute: (3i)^2+9=9i^2+9=9(-1)+9=0.

Q: How do you check x=-3i satisfies the equation?

Substitute: (-3i)^2+9=9i^2+9=-9+9=0.

Q: What is the discriminant method for x^2+9=0?

For ax^2+bx+c=0, a=1,b=0,c=9. Discriminant: b^2-4ac=-36. Then x=\frac{-b\pm \sqrt{-36}}{2}=\pm 3i.

Answer

We solve the quadratic equation \(x^2+9=0\).

Subtract 9 from both sides: \(x^2=-9\).

Take the square roots: \(x=\pm 3i\).

Final result: \(x=\pm 3i\).

Detailed Explanation

We want to solve the equation

\[
x^2 + 9 = 0
\]

Step 1: Isolate the \(x^2\) term.

Subtract \(9\) from both sides so that the \(x^2\) is alone.

\[
x^2 + 9 – 9 = 0 – 9
\]

\[
x^2 = -9
\]

Step 2: Take the square root of both sides.

Now we have \(x^2 = -9\). Taking the square root gives two solutions, using the fact that square roots produce \(\pm\).

\[
x = \pm \sqrt{-9}
\]

Step 3: Rewrite \(\sqrt{-9}\) using imaginary numbers.

We use the identity \(\sqrt{-1} = i\). Also, \(\sqrt{9} = 3\). So:

\[
\sqrt{-9} = \sqrt{9}\sqrt{-1} = 3i
\]

Step 4: Combine with the \(\pm\).

Substitute \(\sqrt{-9} = 3i\) back into the solutions.

\[
x = \pm 3i
\]

Therefore, the solutions are:

\[
x = 3i \quad \text{and} \quad x = -3i
\]

See full solution

Try AI homework help, get step-by-step help now!

Homework AI

Algebra FAQ

How do you solve \(x^2+9=0\)?

Rearrange: \(x^2=-9\). Take square roots: \(x=\pm \sqrt{-9}=\pm 3i\).

Why are the solutions imaginary for \(x^2+9=0\)?

Because \(x^2=-9\), and a real square cannot be negative. So \(x\) must be complex, namely \(x=\pm 3i\).

What does \( \sqrt{-9} \) equal?

Use \( \sqrt{-a}=i\sqrt{a} \). So \( \sqrt{-9}=i\sqrt{9}=3i\).

How do you check \(x=3i\) satisfies the equation?

Substitute: \((3i)^2+9=9i^2+9=9(-1)+9=0\).

How do you check \(x=-3i\) satisfies the equation?

Substitute: \((-3i)^2+9=9i^2+9=-9+9=0\).

What is the discriminant method for \(x^2+9=0\)?

For \(ax^2+bx+c=0\), \(a=1,b=0,c=9\). Discriminant: \(b^2-4ac=-36\). Then \(x=\frac{-b\pm \sqrt{-36}}{2}=\pm 3i\).

Solve x²+9=0 step by step.
Try with smart AI tools now.

298,376+ active customers
Math, Geometry, Trigonometry, etc.

Math Solver Geometry Solver Trigonometry Solver