Finding the Discriminant of the Given Equation

Oct 13, 2025 by wbrum.org 20 views

Understanding the Quadratic Formula

The quadratic formula is a fundamental concept in algebra that allows us to solve quadratic equations of the form $ax^2 + bx + c = 0$ . The formula is given by:

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

The Discriminant

The discriminant, denoted by $\Delta$ or $D$ , is the expression under the square root in the quadratic formula:

\Delta = b^2 - 4ac

The discriminant plays a crucial role in determining the nature of the solutions of a quadratic equation.

Finding the Discriminant of the Given Equation

We are given the quadratic equation $3v^2 + 9v + 6 = 0$ . To find the discriminant, we need to identify the values of $a$ , $b$ , and $c$ .

$a = 3$
$b = 9$
$c = 6$

Now, we can plug these values into the formula for the discriminant:

\Delta = b^2 - 4ac

\Delta = (9)^2 - 4(3)(6)

\Delta = 81 - 72

\Delta = 9

Interpreting the Discriminant

The discriminant is a positive number, which means that the quadratic equation has two distinct real solutions.

Conclusion

In conclusion, the discriminant of the equation $3v^2 + 9v + 6 = 0$ is 9, which indicates that the equation has two real solutions.

Real Solutions of the Equation

To find the real solutions of the equation, we can use the quadratic formula:

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

v = \frac{-9 \pm \sqrt{9^2 - 4(3)(6)}}{2(3)}

v = \frac{-9 \pm \sqrt{81 - 72}}{6}

v = \frac{-9 \pm \sqrt{9}}{6}

v = \frac{-9 \pm 3}{6}

This gives us two possible values for $v$ :

v_1 = \frac{-9 + 3}{6} = \frac{-6}{6} = -1

v_2 = \frac{-9 - 3}{6} = \frac{-12}{6} = -2

Therefore, the equation $3v^2 + 9v + 6 = 0$ has two real solutions: $v = -1$ and $v = -2$ .

Final Answer

The final answer is: C. Two real solutions

Understanding the Quadratic Formula and Discriminant

The quadratic formula is a fundamental concept in algebra that allows us to solve quadratic equations of the form $ax^2 + bx + c = 0$ . The formula is given by:

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

The discriminant, denoted by $\Delta$ or $D$ , is the expression under the square root in the quadratic formula:

\Delta = b^2 - 4ac

The discriminant plays a crucial role in determining the nature of the solutions of a quadratic equation.

Frequently Asked Questions

Q1: What is the discriminant of a quadratic equation?

A1: The discriminant is the expression under the square root in the quadratic formula: $\Delta = b^2 - 4ac$ .

Q2: How do I find the discriminant of a quadratic equation?

A2: To find the discriminant, you need to identify the values of $a$ , $b$ , and $c$ in the quadratic equation. Then, plug these values into the formula for the discriminant: $\Delta = b^2 - 4ac$ .

Q3: What does the discriminant tell us about the solutions of a quadratic equation?

A3: The discriminant tells us about the nature of the solutions of a quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q4: How do I determine the number of real solutions of a quadratic equation?

A4: To determine the number of real solutions of a quadratic equation, you need to find the discriminant. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q5: Can you give an example of finding the discriminant of a quadratic equation?

A5: Let's consider the quadratic equation $3v^2 + 9v + 6 = 0$ . To find the discriminant, we need to identify the values of $a$ , $b$ , and $c$ .

$a = 3$
$b = 9$
$c = 6$

Now, we can plug these values into the formula for the discriminant:

\Delta = b^2 - 4ac

\Delta = (9)^2 - 4(3)(6)

\Delta = 81 - 72

\Delta = 9

The discriminant is 9, which means that the equation has two distinct real solutions.

Q6: Can you explain the quadratic formula in more detail?

A6: The quadratic formula is a fundamental concept in algebra that allows us to solve quadratic equations of the form $ax^2 + bx + c = 0$ . The formula is given by:

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

The quadratic formula is a powerful tool for solving quadratic equations, and it is widely used in mathematics, science, and engineering.

Conclusion

In conclusion, the discriminant is a crucial concept in algebra that helps us determine the nature of the solutions of a quadratic equation. By finding the discriminant, we can determine the number of real solutions of a quadratic equation. We hope that this Q&A article has been helpful in understanding the quadratic formula and discriminant.