Solve the Quadratic Equation: -2x² - 5x - 9 = 0

Q: What does it mean when there's no real solution?

It means the parabola never crosses the x-axis. Since our coefficient of $ x^2 $ is negative (-2), the parabola opens downward and stays completely below the x-axis.

Q: Did I do something wrong if I get no solution?

No! "No solution" is a perfectly valid answer in mathematics. It tells us important information about the quadratic function - that it has no x-intercepts.

Q: How do I know for sure the discriminant is negative?

Calculate step by step: $ b^2 - 4ac = (-5)^2 - 4(-2)(-9) = 25 - 4(18) = 25 - 72 = -47 $. Since -47

Q: What if the problem asked for complex solutions?

Then you'd continue with the quadratic formula using $ \sqrt{-47} = i\sqrt{47} $. But since this asks for real solutions only, "No solution" is the correct answer.

Q: Can I still graph this equation even with no real solutions?

Yes! The graph is a downward-opening parabola that sits entirely below the x-axis. The vertex represents the maximum point of the function.

Quadratic Equations with Negative Discriminant

Solve the following equation:

$-2x^2-5x-9=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find the value of X.

00:11 First, identify the coefficients in the equation.

00:24 Next, use the formula for the roots.

00:41 Substitute the correct values based on the data, and then solve.

01:03 Now, calculate the square and the products.

01:22 Remember, a root cannot be a negative number.

01:36 So, in this case, there is no solution. And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$-2x^2-5x-9=0$

Step-by-step solution

To solve the quadratic equation $-2x^2 - 5x - 9 = 0$ , we follow these steps:

Step 1: Identify coefficients $a = -2$ , $b = -5$ , $c = -9$ .
Step 2: Compute the discriminant, $b^2 - 4ac$ .
Step 3: Analyze the discriminant's value to determine the nature of the roots.

Step 1: We have $a = -2$ , $b = -5$ , $c = -9$ .

Step 2: Calculate the discriminant:
$b^2 - 4ac = (-5)^2 - 4(-2)(-9) = 25 - 72 = -47$ .

Step 3: Since the discriminant value is $-47$ , which is less than zero, this indicates that there are no real solutions because the square root of a negative number is imaginary.

Therefore, since there are no real solutions, we conclude that the problem has no real solutions.

Thus, the solution to the equation is No solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Rule: Calculate discriminant $b^2 - 4ac$ first to determine solution type
Technique: $(-5)^2 - 4(-2)(-9) = 25 - 72 = -47$
Check: Negative discriminant means no real solutions exist ✓

Common Mistakes

Avoid these frequent errors

Calculating the discriminant incorrectly with signs
Don't forget that $-4(-2)(-9) = -72$ not +72! Students often mess up the negative signs and get 97 instead of -47. Always be extra careful with negative coefficients and use parentheses: $b^2 - 4ac = (-5)^2 - 4(-2)(-9)$ .

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's no real solution?

It means the parabola never crosses the x-axis. Since our coefficient of $x^2$ is negative (-2), the parabola opens downward and stays completely below the x-axis.

Did I do something wrong if I get no solution?

No! "No solution" is a perfectly valid answer in mathematics. It tells us important information about the quadratic function - that it has no x-intercepts.

How do I know for sure the discriminant is negative?

Calculate step by step: $b^2 - 4ac = (-5)^2 - 4(-2)(-9) = 25 - 4(18) = 25 - 72 = -47$ . Since -47 < 0, there are no real solutions.

What if the problem asked for complex solutions?

Then you'd continue with the quadratic formula using $\sqrt{-47} = i\sqrt{47}$ . But since this asks for real solutions only, "No solution" is the correct answer.

Can I still graph this equation even with no real solutions?

Yes! The graph is a downward-opening parabola that sits entirely below the x-axis. The vertex represents the maximum point of the function.

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