Quadratic Equation Calculator

Oct 13, 2025 by recoveryshake.com 112 views

Which Equation Has an ${$ a$}$-value of ${ $1\$}$ , a ${$ b$}$-value of ${$ -3$}$, and a ${$ c$}$-value of ${$ -5$}$?

Understanding the Question

To determine which equation has an ${$ a$}$-value of ${ $1\$}$ , a ${$ b$}$-value of ${$ -3$}$, and a ${$ c$}$-value of ${$ -5$}$, we need to understand the structure of a quadratic equation. A quadratic equation is typically written in the form of ${ $0 = ax^2 + bx + c\$}$ , where ${$ a$}$, ${$ b$}$, and ${$ c$}$ are constants.

Analyzing the Options

Let's analyze each option to determine which one matches the given values of ${$ a$}$, ${$ b$}$, and ${$ c$}$.

Option A: ${ $0 = -3x - 5 + x^2\$}$

In this option, the equation is ${ $0 = -3x - 5 + x^2\$}$ . To identify the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$, we need to rewrite the equation in the standard form of a quadratic equation.

Rewriting the Equation

We can rewrite the equation as ${ $0 = x^2 - 3x - 5\$}$ . Comparing this with the standard form of a quadratic equation, we can see that:

${$ a = 1$}$
${$ b = -3$}$
${$ c = -5$}$

This option matches the given values of ${$ a$}$, ${$ b$}$, and ${$ c$}$.

Option B: ${ $0 = x - 3 - 5x^2\$}$

In this option, the equation is ${ $0 = x - 3 - 5x^2\$}$ . To identify the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$, we need to rewrite the equation in the standard form of a quadratic equation.

Rewriting the Equation

We can rewrite the equation as ${ $0 = -5x^2 - x + 3\$}$ . Comparing this with the standard form of a quadratic equation, we can see that:

${$ a = -5$}$
${$ b = -1$}$
${$ c = 3$}$

This option does not match the given values of ${$ a$}$, ${$ b$}$, and ${$ c$}$.

Option C: ${ $0 = 3x - 5\$}$

In this option, the equation is ${ $0 = 3x - 5\$}$ . To identify the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$, we need to rewrite the equation in the standard form of a quadratic equation.

Rewriting the Equation

We can rewrite the equation as ${ $0 = 3x - 5\$}$ . However, this equation is not a quadratic equation, as it does not have a term with ${$ x^2$}$. Therefore, we cannot determine the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ for this option.

Conclusion

Based on the analysis of each option, we can conclude that the equation with an ${$ a$}$-value of ${ $1\$}$ , a ${$ b$}$-value of ${$ -3$}$, and a ${$ c$}$-value of ${$ -5$}$ is:

Option A: ${ $0 = -3x - 5 + x^2\$}$

This equation matches the given values of ${$ a$}$, ${$ b$}$, and ${$ c$}$, and is therefore the correct answer.

Understanding Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and are used to model a wide range of real-world phenomena. A quadratic equation is typically written in the form of ${ $0 = ax^2 + bx + c\$}$ , where ${$ a$}$, ${$ b$}$, and ${$ c$}$ are constants.

Identifying the Values of ${$ a$}$, ${$ b$}$, and ${$ c$}$

To identify the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ in a quadratic equation, we need to rewrite the equation in the standard form of a quadratic equation. This involves rearranging the terms to group the like terms together.

Example

Consider the equation ${ $0 = -3x - 5 + x^2\$}$ . To identify the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$, we need to rewrite the equation in the standard form of a quadratic equation.

Rewriting the Equation

We can rewrite the equation as ${ $0 = x^2 - 3x - 5\$}$ . Comparing this with the standard form of a quadratic equation, we can see that:

${$ a = 1$}$
${$ b = -3$}$
${$ c = -5$}$

This shows that the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ are ${ $1\$}$ , ${$ -3$}$, and ${$ -5$}$, respectively.

Conclusion

In conclusion, the equation with an ${$ a$}$-value of ${ $1\$}$ , a ${$ b$}$-value of ${$ -3$}$, and a ${$ c$}$-value of ${$ -5$}$ is:

Option A: ${ $0 = -3x - 5 + x^2\$}$

This equation matches the given values of ${$ a$}$, ${$ b$}$, and ${$ c$}$, and is therefore the correct answer.
Quadratic Equation Q&A

Understanding Quadratic Equations

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually ${$ x$}$) is two. It is typically written in the form of ${ $0 = ax^2 + bx + c\$}$ , where ${$ a$}$, ${$ b$}$, and ${$ c$}$ are constants.

Q: How do I identify the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ in a quadratic equation?

A: To identify the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ in a quadratic equation, you need to rewrite the equation in the standard form of a quadratic equation. This involves rearranging the terms to group the like terms together.

Q: What is the standard form of a quadratic equation?

A: The standard form of a quadratic equation is ${ $0 = ax^2 + bx + c\$}$ , where ${$ a$}$, ${$ b$}$, and ${$ c$}$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use various methods such as factoring, completing the square, or using the quadratic formula. The quadratic formula is ${$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$}$, where ${$ a$}$, ${$ b$}$, and ${$ c$}$ are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is ${$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$}$, where ${$ a$}$, ${$ b$}$, and ${$ c$}$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of ${$ a$}$, ${$ b$}$, and ${$ c$}$ into the formula and simplify.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a term with ${$ x^2$}$, while a linear equation does not have a term with ${$ x^2$}$.

Q: Can I use the quadratic formula to solve a linear equation?

A: No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is used to solve quadratic equations, not linear equations.

Q: Can I use the quadratic formula to solve a cubic equation?

A: No, you cannot use the quadratic formula to solve a cubic equation. The quadratic formula is used to solve quadratic equations, not cubic equations.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and are used to model a wide range of real-world phenomena. The quadratic formula is a powerful tool for solving quadratic equations, and is widely used in mathematics and science. By understanding the basics of quadratic equations and the quadratic formula, you can solve a wide range of problems and applications.

Additional Resources

Quadratic Equation Calculator: A calculator that can be used to solve quadratic equations.
Quadratic Formula Derivation: A derivation of the quadratic formula.
Quadratic Equation Examples: Examples of quadratic equations and their solutions.
Quadratic Equation Applications: Applications of quadratic equations in mathematics and science.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and are used to model a wide range of real-world phenomena. By understanding the basics of quadratic equations and the quadratic formula, you can solve a wide range of problems and applications. Whether you are a student, a teacher, or a professional, quadratic equations are an essential tool for solving problems and making predictions.