Solve the Quadratic Equation: 3x² - 9x + 12 = 0

Q: What does it mean when the discriminant is negative?

A negative discriminant means the parabola doesn't cross the x-axis at all! Since we can only see real solutions as x-intercepts, this equation has no real solutions.

Q: Can I still solve it if the discriminant is negative?

Technically yes, but the solutions would be complex numbers involving the imaginary unit i. For most algebra courses, we simply say "no real solutions" or "no solution."

Q: How do I know if I calculated the discriminant correctly?

Double-check your arithmetic: $ \Delta = b^2 - 4ac $. For this problem: $ (-9)^2 = 81 $, $ 4 \times 3 \times 12 = 144 $, so $ \Delta = 81 - 144 = -63 $.

Q: Why can't we have square roots of negative numbers?

In the real number system, we can't take square roots of negative numbers because no real number times itself gives a negative result. That's why negative discriminants mean no real solutions!

Q: What if I made an error and the discriminant should be positive?

Always double-check your coefficients from the original equation! Make sure $ a = 3 $, $ b = -9 $, and $ c = 12 $ are correct before calculating.

Quadratic Equations with Negative Discriminant

Solve the equation

$3x^2-9x+12=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Alright! Let's find the value of X.

00:09 First, notice the coefficients. They're important.

00:16 Next, we'll use the root formula. It's a handy tool!

00:29 Now, plug in the given values. And solve for X, step by step.

00:49 Time to calculate the squares and products. Almost there!

01:05 Oops! Remember, you can't have a root of a negative number. So, n o solution here.

01:11 And there we have it! That's the answer to our problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the equation

$3x^2-9x+12=0$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the coefficients $a = 3$ , $b = -9$ , $c = 12$ .
Step 2: Compute the discriminant $\Delta = b^2 - 4ac$ .
Step 3: Determine if real solutions exist based on the discriminant.
Step 4: Solve using the Quadratic Formula if applicable.

Now, let's work through each step:

Step 2: Calculate the discriminant:

$\Delta = (-9)^2 - 4 \times 3 \times 12$

$\Delta = 81 - 144 = -63$

Step 3: Since the discriminant $\Delta$ is negative ( $-63$ ), this means there are no real solutions for the quadratic equation. A negative discriminant indicates that the solutions are complex (i.e., non-real), so the equation has no real solution.

Therefore, the solution to the problem is No solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $\Delta = b^2 - 4ac < 0$ , no real solutions exist
Calculation: $\Delta = (-9)^2 - 4(3)(12) = 81 - 144 = -63$
Check: Negative discriminant confirms no real solutions for the equation ✓

Common Mistakes

Avoid these frequent errors

Ignoring the discriminant and attempting to use quadratic formula anyway
Don't calculate $x = \frac{-b \pm \sqrt{-63}}{2a}$ when discriminant is negative = imaginary numbers! This leads to complex solutions, not real ones. Always check the discriminant first to determine if real solutions exist.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ 2x^2-10x-12=0 $$

FAQ

Everything you need to know about this question

What does it mean when the discriminant is negative?

A negative discriminant means the parabola doesn't cross the x-axis at all! Since we can only see real solutions as x-intercepts, this equation has no real solutions.

Can I still solve it if the discriminant is negative?

Technically yes, but the solutions would be complex numbers involving the imaginary unit i. For most algebra courses, we simply say "no real solutions" or "no solution."

How do I know if I calculated the discriminant correctly?

Double-check your arithmetic: $\Delta = b^2 - 4ac$ . For this problem: $(-9)^2 = 81$ , $4 \times 3 \times 12 = 144$ , so $\Delta = 81 - 144 = -63$ .

Why can't we have square roots of negative numbers?

In the real number system, we can't take square roots of negative numbers because no real number times itself gives a negative result. That's why negative discriminants mean no real solutions!

What if I made an error and the discriminant should be positive?

Always double-check your coefficients from the original equation! Make sure $a = 3$ , $b = -9$ , and $c = 12$ are correct before calculating.

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