Solve the Fractional Quadratic: x²/2 + x/4 + 1 = 0

Q: What does it mean when there are no real solutions?

It means the parabola doesn't cross the x-axis! When discriminant , the quadratic has no x-intercepts, so there are no real values of x that satisfy the equation.

Q: Should I still try to use the quadratic formula?

You could, but you'd get square roots of negative numbers (imaginary solutions). For most problems at this level, "No solution" is the expected answer when discriminant is negative.

Q: How do I clear fractions before finding the discriminant?

Multiply every term by the LCD of all denominators. Here, LCD of 2 and 4 is 4, so: $ 4 \cdot \frac{x^2}{2} + 4 \cdot \frac{x}{4} + 4 \cdot 1 = 2x^2 + x + 4 $

Q: Can a quadratic equation ever have just one solution?

Yes! When the discriminant equals exactly zero, there's one repeated real solution. When discriminant > 0, there are two different real solutions.

Q: Why is the discriminant so important?

The discriminant $ b^2 - 4ac $ tells you everything about solutions before you calculate them:Negative: No real solutionsZero: One solutionPositive: Two solutions

Quadratic Equations with Negative Discriminant

Solve the following equation:

$\frac{x^2}{2}+\frac{x}{4}+1=0$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Multiply by 4 to eliminate fractions

00:19 Identify the coefficients

00:25 Use the roots formula

00:44 Substitute appropriate values according to the given data and solve

01:09 Calculate the square and products

01:18 There's no such thing as a root of a negative number

01:31 Therefore there is no solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$\frac{x^2}{2}+\frac{x}{4}+1=0$

Step-by-step solution

To solve the quadratic equation $\frac{x^2}{2} + \frac{x}{4} + 1 = 0$ , follow these steps:

Step 1: Convert to Standard Form
Begin by eliminating the fractions to simplify. Multiply the entire equation by 4, the least common multiple of the denominators: $4 \left( \frac{x^2}{2} + \frac{x}{4} + 1 \right) = 4 \cdot 0$ which simplifies to: $2x^2 + x + 4 = 0$ Now, it's in standard quadratic form: $ax^2 + bx + c = 0$ , with $a = 2$ , $b = 1$ , and $c = 4$ .
Step 2: Evaluate the Discriminant
The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is calculated as: $b^2 - 4ac$ Substituting the values, we have: $1^2 - 4 \cdot 2 \cdot 4 = 1 - 32 = -31$ Since the discriminant is negative, it indicates that there are no real solutions for the equation.

Conclusion: The given quadratic equation has no real solutions due to the negative discriminant.

The correct answer to the problem is therefore, No solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Standard Form: Multiply by LCD to eliminate fractions first
Discriminant: Calculate $b^2 - 4ac = 1 - 32 = -31$
Check: Negative discriminant means no real solutions exist ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check the discriminant before solving
Don't jump straight to the quadratic formula without checking $b^2 - 4ac$ first = wasted time on impossible calculations! When the discriminant is negative, there are no real solutions. Always evaluate the discriminant to determine if real solutions exist.

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

What does it mean when there are no real solutions?

It means the parabola doesn't cross the x-axis! When discriminant < 0, the quadratic has no x-intercepts, so there are no real values of x that satisfy the equation.

Should I still try to use the quadratic formula?

You could, but you'd get square roots of negative numbers (imaginary solutions). For most problems at this level, "No solution" is the expected answer when discriminant is negative.

How do I clear fractions before finding the discriminant?

Multiply every term by the LCD of all denominators. Here, LCD of 2 and 4 is 4, so: $4 \cdot \frac{x^2}{2} + 4 \cdot \frac{x}{4} + 4 \cdot 1 = 2x^2 + x + 4$

Can a quadratic equation ever have just one solution?

Yes! When the discriminant equals exactly zero, there's one repeated real solution. When discriminant > 0, there are two different real solutions.

Why is the discriminant so important?

The discriminant $b^2 - 4ac$ tells you everything about solutions before you calculate them:

Negative: No real solutions
Zero: One solution
Positive: Two solutions

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Solving Quadratic Equations questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime