Solve the Quadratic Equation: -5x² - 2x - 8 = 0

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

It means the parabola never touches or crosses the x-axis. Since \( a = -5 , this parabola opens downward and stays completely below the x-axis.

Q: Can I still get an answer using complex numbers?

Yes! In advanced math, you'd get $ x = \frac{2 \pm i\sqrt{156}}{-10} $, but for this level, "no solution" is the correct answer when working with real numbers only.

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

Always calculate it first when solving quadratic equations! It tells you how many real solutions to expect: positive = 2 solutions, zero = 1 solution, negative = no real solutions.

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

Double-check: $ D = b^2 - 4ac = (-2)^2 - 4(-5)(-8) $. Remember that negative times negative equals positive, so $ 4(-5)(-8) = +160 $.

Q: What if I tried factoring instead?

You can try, but $ -5x^2 - 2x - 8 = 0 $ doesn't factor with real numbers. The negative discriminant confirms that no factoring method will work for real solutions.

Quadratic Equations with Negative Discriminant

$-5x^2-2x-8=0$

Solve the equation

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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

00:09 First, focus on the coefficients. They're very important.

00:16 Next, we'll use the roots formula. Step by step, I'll guide you.

00:29 Now, substitute the given values in the formula. Let's solve for X together.

00:57 Calculate the products and then find the square.

01:09 If we encounter a negative number under a root, there's no real solution.

01:16 And that's how you find the solution to this question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$-5x^2-2x-8=0$

Solve the equation

Step-by-step solution

Let's solve the quadratic equation $-5x^2 - 2x - 8 = 0$ using the quadratic formula:

First, identify the coefficients: $a = -5$ , $b = -2$ , and $c = -8$ .

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

We start by calculating the discriminant, $D = b^2 - 4ac$ :

$D = (-2)^2 - 4 \cdot (-5) \cdot (-8)$

$D = 4 - 160$

$D = -156$

Since the discriminant $D = -156$ is less than zero, this indicates that there are no real solutions for the quadratic equation $-5x^2 - 2x - 8 = 0$ .

Therefore, the equation has no solution in terms of real numbers.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $b^2 - 4ac < 0$ , no real solutions exist
Calculation: $D = (-2)^2 - 4(-5)(-8) = 4 - 160 = -156$
Check: Negative discriminant means parabola doesn't cross x-axis ✓

Common Mistakes

Avoid these frequent errors

Continuing to use quadratic formula with negative discriminant
Don't try to find $\sqrt{-156}$ in real numbers = impossible result! The negative discriminant tells you to stop because no real solutions exist. Always check the discriminant first and conclude 'no solution' when 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 there are no real solutions?

It means the parabola never touches or crosses the x-axis. Since $a = -5 < 0$ , this parabola opens downward and stays completely below the x-axis.

Can I still get an answer using complex numbers?

Yes! In advanced math, you'd get $x = \frac{2 \pm i\sqrt{156}}{-10}$ , but for this level, "no solution" is the correct answer when working with real numbers only.

How do I know when to calculate the discriminant?

Always calculate it first when solving quadratic equations! It tells you how many real solutions to expect: positive = 2 solutions, zero = 1 solution, negative = no real solutions.

Could I have made an error in my discriminant calculation?

Double-check: $D = b^2 - 4ac = (-2)^2 - 4(-5)(-8)$ . Remember that negative times negative equals positive, so $4(-5)(-8) = +160$ .

What if I tried factoring instead?

You can try, but $-5x^2 - 2x - 8 = 0$ doesn't factor with real numbers. The negative discriminant confirms that no factoring method will work for real solutions.

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