Solve the Quadratic: Unravel the Equation 5x² + 1.5x + 1 = 0

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

A negative discriminant means the parabola doesn't cross the x-axis, so there are no real solutions. The equation has complex solutions instead, which you'll learn about in advanced algebra.

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

Double-check each step: $ b^2 = (1.5)^2 = 2.25 $, then $ 4ac = 4(5)(1) = 20 $, so $ \Delta = 2.25 - 20 = -17.75 $. The negative result is correct!

Q: Can I still graph this quadratic equation?

Absolutely! The parabola still exists - it just opens upward and sits entirely above the x-axis. You can find the vertex and other points normally.

Q: Why does the problem ask for 'no solution' instead of complex solutions?

In basic algebra, we focus on real solutions that you can plot on a regular number line. Complex solutions involving imaginary numbers are typically covered in advanced courses.

Q: What would happen if the discriminant were zero?

If $ \Delta = 0 $, you'd have exactly one real solution (a repeated root). The parabola would just touch the x-axis at one point called the vertex.

Quadratic Equations with Negative Discriminant

Solve the following equation:

$5x^2+1\frac{1}{2}x+1=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's find the value of X.

00:12 First, we need to identify the coefficients. Pause and look at the equation. Ready?

00:28 Now, we'll use the roots formula. Take it step by step and follow along.

00:49 Let's substitute the appropriate values given in the problem, and solve it together.

01:14 Next, we calculate the square and the products. Let's do it slowly.

01:36 Remember, you can't have a root of a negative number in this case.

01:40 Therefore, there is no solution to this problem. And that's okay!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$5x^2+1\frac{1}{2}x+1=0$

Step-by-step solution

Let's solve the quadratic equation $5x^2 + 1.5x + 1 = 0$ using the quadratic formula.

Step 1: Identify the coefficients for the equation $ax^2 + bx + c = 0$ :
$a = 5$
$b = 1.5$
$c = 1$

Step 2: Calculate the discriminant $\Delta = b^2 - 4ac$ .
$\Delta = (1.5)^2 - 4 \cdot 5 \cdot 1 = 2.25 - 20 = -17.75$

Step 3: Analyze the discriminant:
Since the discriminant $\Delta = -17.75$ is negative, this indicates that there are no real solutions to the equation $5x^2 + 1.5x + 1 = 0$ .

Therefore, the equation has no solution in the real number system.

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: When $\Delta = -17.75 < 0$ , no real solutions exist
Check: Verify discriminant calculation: $(1.5)^2 - 4(5)(1) = -17.75$ ✓

Common Mistakes

Avoid these frequent errors

Forcing real solutions when discriminant is negative
Don't continue with the quadratic formula when $\Delta < 0$ = impossible square root of negative number! This creates mathematical errors and confusion. Always check the discriminant first and conclude 'no real solutions' 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 the discriminant is negative?

A negative discriminant means the parabola doesn't cross the x-axis, so there are no real solutions. The equation has complex solutions instead, which you'll learn about in advanced algebra.

How do I know if I calculated the discriminant correctly?

Double-check each step: $b^2 = (1.5)^2 = 2.25$ , then $4ac = 4(5)(1) = 20$ , so $\Delta = 2.25 - 20 = -17.75$ . The negative result is correct!

Can I still graph this quadratic equation?

Absolutely! The parabola still exists - it just opens upward and sits entirely above the x-axis. You can find the vertex and other points normally.

Why does the problem ask for 'no solution' instead of complex solutions?

In basic algebra, we focus on real solutions that you can plot on a regular number line. Complex solutions involving imaginary numbers are typically covered in advanced courses.

What would happen if the discriminant were zero?

If $\Delta = 0$ , you'd have exactly one real solution (a repeated root). The parabola would just touch the x-axis at one point called the vertex.

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