Understanding the Roots of a Quadratic Equation: 3x2 - 7x - 10 0

Understanding the nature of the roots of a quadratic equation is a critical concept in algebra. This article will explore how we can determine whether the quadratic equation 3x2 - 7x - 10 0 has real roots, purely imaginary roots, or neither. We will employ the quadratic formula and the concept of the discriminant to find the solution.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, generally expressed in the form:

ax2 bx c 0

where a, b, and c are constants and a ≠ 0. This article focuses on the quadratic equation 3x2 - 7x - 10 0.

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations and is given by:

x frac{-b pm sqrt{b2 - 4ac}}{2a}

Applying the formula to the equation 3x2 - 7x - 10 0 where a 3, b -7, and c -10, we get:

x frac{-(-7) pm sqrt{(-7)2 - 4 cdot 3 cdot (-10)}}{2 cdot 3}

Let's simplify the discriminant first:

D (-7)2 - 4 cdot 3 cdot (-10) 49 120 169

Now, we can substitute this back into the quadratic formula:

x frac{7 pm sqrt{169}}{6}

Since the discriminant evaluates to 169, which is a positive number, we find the roots:

x frac{7 pm 13}{6}

Therefore, the roots are:

x frac{7 13}{6} frac{20}{6} frac{10}{3}

x frac{7 - 13}{6} frac{-6}{6} -1

This example demonstrates that this particular quadratic equation has real roots.

Exploring Complex Roots: When the Discriminant is Negative

Create an interesting example and guide the reader to understand the concept of complex roots. The example provided initially has a negative discriminant, which we will now explore.

For the quadratic equation:

3x2 - 7x - 10 0

Using the quadratic formula:

x frac{-b pm sqrt{b2 - 4ac}}{2a}

where a 3, b -7, and c -10, the discriminant is:

D b2 - 4ac

D (-7)2 - 4 cdot 3 cdot (-10) 49 120 169

Given that the discriminant is 169, the roots are real. However, if we modify this to a different equation, we can show how it can lead to complex roots.

Consider the equation:

3x2 - 7x - 10 -20

This simplifies to:

3x2 - 7x 10 0

b2 - 4ac (-7)2 - 4 cdot 3 cdot 10 49 - 120 -71

Since the discriminant is -71, which is negative, the roots are complex:

x frac{-(-7) pm sqrt{-71}}{2 cdot 3}

x frac{7 pm isqrt{71}}{6}

This indicates that the roots are of the form:

x -frac{7}{6} pm frac{isqrt{71}}{6}

Using the Discriminant to Analyze the Roots

The discriminant b2 - 4ac is a key factor in determining the nature of the roots of a quadratic equation:

Positive Discriminant: The equation has two real and distinct roots. Zero Discriminant: The equation has one real root (a repeated root). Negative Discriminant: The equation has two complex conjugate roots.

For the equation:

3x2 - 7x - 10 0

The discriminant is:

D b2 - 4ac (-7)2 - 4 cdot 3 cdot (-10) 49 120 169

Since the discriminant is positive and not zero, the equation has two real and distinct roots.

Conclusion

In summary, we have explored the process of determining the nature of the roots of a quadratic equation using the quadratic formula and the discriminant. The roots of the quadratic equation 3x2 - 7x - 10 0 are real and distinct, while the equation 3x2 - 7x - 10 -20 has two complex conjugate roots. Understanding these concepts is crucial for solving problems in algebra and related fields.

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