Substituting Values into the Quadratic Formula

Oct 13, 2025 by miodeya.com 112 views

Introduction

The quadratic formula is a powerful tool used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . It is a fundamental concept in algebra and is widely used in various fields such as physics, engineering, and economics. In this article, we will explore the correct substitution of values from the equation $0 = -3x^2 - 2x + 6$ into the quadratic formula.

The Quadratic Formula

The quadratic 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.

Substituting Values into the Quadratic Formula

To solve the equation $0 = -3x^2 - 2x + 6$ , we need to substitute the values of $a$ , $b$ , and $c$ into the quadratic formula. The values are:

$a = -3$
$b = -2$
$c = 6$

Substituting these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

x = \frac{2 \pm \sqrt{4 + 72}}{-6}

x = \frac{2 \pm \sqrt{76}}{-6}

x = \frac{2 \pm 2\sqrt{19}}{-6}

Simplifying the Expression

To simplify the expression further, we can divide both the numerator and the denominator by $-2$ . This gives us:

x = \frac{1 \pm \sqrt{19}}{-3}

Conclusion

In this article, we have shown the correct substitution of values from the equation $0 = -3x^2 - 2x + 6$ into the quadratic formula. We have also simplified the expression to obtain the final solution. The quadratic formula is a powerful tool used to solve quadratic equations, and it is essential to understand how to substitute values into the formula correctly.

Common Mistakes to Avoid

When substituting values into the quadratic formula, it is essential to avoid common mistakes such as:

Swapping the values of $a$ and $b$
Forgetting to include the $\pm$ symbol
Not simplifying the expression correctly

By avoiding these common mistakes, you can ensure that you obtain the correct solution to the quadratic equation.

Real-World Applications

The quadratic formula has numerous real-world applications in various fields such as:

Physics: The quadratic formula is used to describe the motion of objects under the influence of gravity.
Engineering: The quadratic formula is used to design and optimize systems such as bridges and buildings.
Economics: The quadratic formula is used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, the quadratic formula is a powerful tool used to solve quadratic equations. By understanding how to substitute values into the formula correctly, you can obtain the final solution to the equation. The quadratic formula has numerous real-world applications, and it is essential to understand its significance in various fields.

Final Thoughts

The quadratic formula is a fundamental concept in algebra, and it is essential to understand how to substitute values into the formula correctly. By avoiding common mistakes and simplifying the expression correctly, you can ensure that you obtain the correct solution to the quadratic equation. The quadratic formula has numerous real-world applications, and it is a valuable tool in various fields.

References

[1] "Quadratic Formula" by Math Is Fun
[2] "Quadratic Equations" by Khan Academy
[3] "Quadratic Formula" by Wolfram MathWorld

Additional Resources

For further learning, you can explore the following resources:

Khan Academy: Quadratic Equations
Math Is Fun: Quadratic Formula
Wolfram MathWorld: Quadratic Formula

FAQs

Q: What is the quadratic formula? A: The quadratic formula is a powerful tool used to solve quadratic equations of the form $ax^2 + bx + c = 0$ .

Q: How do I substitute values into the quadratic formula? A: To substitute values into the quadratic formula, you need to identify the values of $a$ , $b$ , and $c$ in the equation and plug them into the formula.

Q: What are the common mistakes to avoid when substituting values into the quadratic formula? A: Common mistakes to avoid include swapping the values of $a$ and $b$ , forgetting to include the $\pm$ symbol, and not simplifying the expression correctly.

Frequently Asked Questions

Q: What is the quadratic formula? A: The quadratic formula is a powerful tool used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . It is given by:

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

Q: How do I use the quadratic formula to solve a quadratic equation? A: To use the quadratic formula, you need to:

Identify the values of $a$ , $b$ , and $c$ in the equation.
Plug these values into the quadratic formula.
Simplify the expression to obtain the final solution.

Q: What are the common mistakes to avoid when using the quadratic formula? A: Common mistakes to avoid include:

Swapping the values of $a$ and $b$
Forgetting to include the $\pm$ symbol
Not simplifying the expression correctly

Q: Can I use the quadratic formula to solve all types of quadratic equations? A: Yes, the quadratic formula can be used to solve all types of quadratic equations, including:

Equations with real roots
Equations with complex roots
Equations with repeated roots

Q: How do I determine the number of solutions to a quadratic equation? A: To determine the number of solutions to a quadratic equation, you need to:

Calculate the discriminant ( $b^2 - 4ac$ )
If the discriminant is positive, the equation has two distinct real solutions.
If the discriminant is zero, the equation has one repeated real solution.
If the discriminant is negative, the equation has two complex solutions.

Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients? A: Yes, the quadratic formula can be used to solve quadratic equations with complex coefficients. However, you need to be careful when simplifying the expression to obtain the final solution.

Q: How do I apply the quadratic formula in real-world problems? A: The quadratic formula has numerous real-world applications in various fields such as:

Physics: The quadratic formula is used to describe the motion of objects under the influence of gravity.
Engineering: The quadratic formula is used to design and optimize systems such as bridges and buildings.
Economics: The quadratic formula is used to model economic systems and make predictions about future trends.

Q: Can I use the quadratic formula to solve quadratic equations with fractional coefficients? A: Yes, the quadratic formula can be used to solve quadratic equations with fractional coefficients. However, you need to be careful when simplifying the expression to obtain the final solution.

Q: How do I determine the nature of the solutions to a quadratic equation? A: To determine the nature of the solutions to a quadratic equation, you need to:

Calculate the discriminant ( $b^2 - 4ac$ )
If the discriminant is positive, the equation has two distinct real solutions.
If the discriminant is zero, the equation has one repeated real solution.
If the discriminant is negative, the equation has two complex solutions.

Q: Can I use the quadratic formula to solve quadratic equations with irrational coefficients? A: Yes, the quadratic formula can be used to solve quadratic equations with irrational coefficients. However, you need to be careful when simplifying the expression to obtain the final solution.

Conclusion

The quadratic formula is a powerful tool used to solve quadratic equations. By understanding how to use the quadratic formula correctly, you can solve a wide range of quadratic equations and apply the formula in real-world problems. Remember to avoid common mistakes and simplify the expression correctly to obtain the final solution.

Additional Resources

For further learning, you can explore the following resources:

Khan Academy: Quadratic Equations
Math Is Fun: Quadratic Formula
Wolfram MathWorld: Quadratic Formula

FAQs

Q: What is the quadratic formula? A: The quadratic formula is a powerful tool used to solve quadratic equations of the form $ax^2 + bx + c = 0$ .

Q: How do I use the quadratic formula to solve a quadratic equation? A: To use the quadratic formula, you need to:

Identify the values of $a$ , $b$ , and $c$ in the equation.
Plug these values into the quadratic formula.
Simplify the expression to obtain the final solution.

Q: What are the common mistakes to avoid when using the quadratic formula? A: Common mistakes to avoid include:

Swapping the values of $a$ and $b$
Forgetting to include the $\pm$ symbol
Not simplifying the expression correctly

Q: Can I use the quadratic formula to solve all types of quadratic equations? A: Yes, the quadratic formula can be used to solve all types of quadratic equations, including:

Equations with real roots
Equations with complex roots
Equations with repeated roots

Q: How do I determine the number of solutions to a quadratic equation? A: To determine the number of solutions to a quadratic equation, you need to:

Calculate the discriminant ( $b^2 - 4ac$ )
If the discriminant is positive, the equation has two distinct real solutions.
If the discriminant is zero, the equation has one repeated real solution.
If the discriminant is negative, the equation has two complex solutions.

Q: How do I apply the quadratic formula in real-world problems? A: The quadratic formula has numerous real-world applications in various fields such as:

Physics: The quadratic formula is used to describe the motion of objects under the influence of gravity.
Engineering: The quadratic formula is used to design and optimize systems such as bridges and buildings.
Economics: The quadratic formula is used to model economic systems and make predictions about future trends.

Q: How do I determine the nature of the solutions to a quadratic equation? A: To determine the nature of the solutions to a quadratic equation, you need to:

Calculate the discriminant ( $b^2 - 4ac$ )
If the discriminant is positive, the equation has two distinct real solutions.
If the discriminant is zero, the equation has one repeated real solution.
If the discriminant is negative, the equation has two complex solutions.