Solve the Quadratic Equation Featuring a Fraction: x² + 3x + 2.5 = 0

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

When \( b^2 - 4ac , the quadratic equation has no real solutions. The parabola doesn't cross the x-axis, so there are no x-intercepts.

Q: Why can't I just ignore the negative sign under the square root?

You cannot take the square root of a negative number in real numbers. While complex numbers exist, this problem asks for real solutions only, so the answer is no solution.

Q: How can I tell beforehand if a quadratic has no real solutions?

Calculate the discriminant $ \Delta = b^2 - 4ac $ first! If it's negative, stop there - no real solutions exist. If it's zero or positive, proceed with the quadratic formula.

Q: Could I have made an error in my discriminant calculation?

Always double-check: $ a = 1 $, $ b = 3 $, $ c = 2.5 $. So $ \Delta = 3^2 - 4(1)(2.5) = 9 - 10 = -1 $. Negative discriminant confirms no real solutions.

Q: What would the graph of this equation look like?

The parabola $ y = x^2 + 3x + 2.5 $ opens upward but stays above the x-axis. It never touches or crosses the x-axis, which is why there are no real solutions.

Quadratic Equations with Negative Discriminant

Solve the following equation:

$x^2+3x+2\frac{1}{2}=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Let's identify the coefficients

00:18 Let's use the roots formula

00:37 Let's substitute appropriate values according to the given data and solve

00:59 Let's calculate the square and products

01:22 There is no such thing as a root of a negative number

01:32 Therefore there is no solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$x^2+3x+2\frac{1}{2}=0$

Step-by-step solution

To solve this problem, we'll apply the quadratic formula. The steps are as follows:

Identify the coefficients: $a = 1$ , $b = 3$ , $c = 2.5$ .
Calculate the discriminant, $\Delta = b^2 - 4ac$ .
Apply the quadratic formula if the discriminant is non-negative.

Let's evaluate the discriminant:
Discriminant, $\Delta = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 2.5 = 9 - 10 = -1$ .

The discriminant is $-1$ , which is less than zero. This means the quadratic equation has no real solutions. Complex solutions are not considered here based on the problem context.

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: Calculate $b^2 - 4ac$ to determine solution type
Technique: For $x^2 + 3x + 2.5 = 0$ , discriminant = $9 - 10 = -1$
Check: If discriminant < 0, equation has no real solutions ✓

Common Mistakes

Avoid these frequent errors

Continuing with quadratic formula when discriminant is negative
Don't apply $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ when discriminant = -1 = undefined square root! This creates imaginary numbers that aren't real solutions. Always check the discriminant first and stop if it's negative.

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 the discriminant is negative?

When $b^2 - 4ac < 0$ , the quadratic equation has no real solutions. The parabola doesn't cross the x-axis, so there are no x-intercepts.

Why can't I just ignore the negative sign under the square root?

You cannot take the square root of a negative number in real numbers. While complex numbers exist, this problem asks for real solutions only, so the answer is no solution.

How can I tell beforehand if a quadratic has no real solutions?

Calculate the discriminant $\Delta = b^2 - 4ac$ first! If it's negative, stop there - no real solutions exist. If it's zero or positive, proceed with the quadratic formula.

Could I have made an error in my discriminant calculation?

Always double-check: $a = 1$ , $b = 3$ , $c = 2.5$ . So $\Delta = 3^2 - 4(1)(2.5) = 9 - 10 = -1$ . Negative discriminant confirms no real solutions.

What would the graph of this equation look like?

The parabola $y = x^2 + 3x + 2.5$ opens upward but stays above the x-axis. It never touches or crosses the x-axis, which is why there are no real solutions.

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