Solve the Equation: x² + 9 = 0 Step-by-Step

Q: Why can't I just say x = ±3 when x² = -9?

Because no real number squared gives a negative result. When you square any real number (positive or negative), you always get a positive result or zero. Since $ 3^2 = 9 $ and $ (-3)^2 = 9 $, neither equals -9.

Q: What does 'no solution' actually mean?

It means there are no real numbers that satisfy the equation. The equation has no solutions in the real number system we typically work with in algebra.

Q: How do I know when to use the discriminant?

Always calculate the discriminant $ b^2 - 4ac $ for quadratic equations. If it's negative, you have no real solutions. If it's zero, you have one solution. If it's positive, you have two solutions.

Q: Are there any solutions at all?

In advanced mathematics, there are complex solutions involving imaginary numbers ($ x = ±3i $), but in basic algebra, we say 'no solution' when referring to real numbers only.

Q: Could I have made a calculation error?

Always double-check! Verify $ a = 1 $, $ b = 0 $, $ c = 9 $, then calculate $ 0^2 - 4(1)(9) = -36 $. Since this is negative, 'no solution' is correct.

Quadratic Equations with Complex Solutions

$x^2+9=0$

Solve the equation

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Isolate X

00:08 Every number squared is greater than 0

00:14 Minus 9 is negative, therefore there is no solution

00:18 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$x^2+9=0$

Solve the equation

Step-by-step solution

The parameters are expressed in the quadratic equation as follows:

aX2+bX+c=0

We identify that we have:
a=1
b=0
c=9

We recall the root formula:

$Roots formula | The version$

We replace according to the formula:

-0 ± √(0²-4*1*9)

We will focus on the part inside the square root (also called delta)

√(0-4*1*9)

√(0-36)

√-36

It is not possible to take the square root of a negative number.

And so the question has no solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $b^2 - 4ac < 0$ , equation has no real solutions
Technique: Calculate discriminant: $0^2 - 4(1)(9) = -36$ (negative)
Check: Negative discriminant means $\sqrt{-36}$ is undefined in real numbers ✓

Common Mistakes

Avoid these frequent errors

Attempting to solve by isolating x²
Don't subtract 9 from both sides to get $x^2 = -9$ then take the square root = $x = ±3$ ! This ignores that square roots of negative numbers don't exist in real numbers. Always check if the discriminant is negative before attempting to find real solutions.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

Why can't I just say x = ±3 when x² = -9?

Because no real number squared gives a negative result. When you square any real number (positive or negative), you always get a positive result or zero. Since $3^2 = 9$ and $(-3)^2 = 9$ , neither equals -9.

What does 'no solution' actually mean?

It means there are no real numbers that satisfy the equation. The equation has no solutions in the real number system we typically work with in algebra.

How do I know when to use the discriminant?

Always calculate the discriminant $b^2 - 4ac$ for quadratic equations. If it's negative, you have no real solutions. If it's zero, you have one solution. If it's positive, you have two solutions.

Are there any solutions at all?

In advanced mathematics, there are complex solutions involving imaginary numbers ( $x = ±3i$ ), but in basic algebra, we say 'no solution' when referring to real numbers only.

Could I have made a calculation error?

Always double-check! Verify $a = 1$ , $b = 0$ , $c = 9$ , then calculate $0^2 - 4(1)(9) = -36$ . Since this is negative, 'no solution' is correct.

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