What Is The Discriminant Of The Quadratic Equation $-9x^2 + 7x - 7 = 0$?A. $-301$ B. $301$ C. $203$ D. $-203$

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Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this case, we have the quadratic equation $-9x^2 + 7x - 7 = 0$.

The Discriminant of a Quadratic Equation

The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol $\Delta$ or $D$. The discriminant is used to determine the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Calculating the Discriminant

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by the formula $\Delta = b^2 - 4ac$. In this case, we have $a = -9$, $b = 7$, and $c = -7$. Plugging these values into the formula, we get:

$$\Delta = (7)^2 - 4(-9)(-7)$$

Simplifying the Expression

Simplifying the expression, we get:

$$\Delta = 49 - 252$$

Evaluating the Expression

Evaluating the expression, we get:

$$\Delta = -203$$

Conclusion

Therefore, the discriminant of the quadratic equation $-9x^2 + 7x - 7 = 0$ is $-203$.

Why is the Discriminant Important?

The discriminant is an important concept in algebra because it helps us determine the nature of the roots of a quadratic equation. If the discriminant is positive, we can use the quadratic formula to find the roots of the equation. If the discriminant is zero, we can use the fact that the equation has one real root to find the root. If the discriminant is negative, we can use the fact that the equation has no real roots to conclude that the equation has complex roots.

Real-World Applications of the Discriminant

The discriminant has many real-world applications. For example, in physics, the discriminant is used to determine the stability of a system. In engineering, the discriminant is used to determine the safety of a structure. In economics, the discriminant is used to determine the competitiveness of a market.

Common Mistakes to Avoid

When calculating the discriminant, there are several common mistakes to avoid. One mistake is to forget to square the value of $b$. Another mistake is to forget to multiply the value of $a$ by the value of $c$. A third mistake is to forget to subtract the product of $4ac$ from the square of $b$.

Tips for Calculating the Discriminant

To calculate the discriminant, follow these tips:

• Make sure to square the value of $b$.
• Make sure to multiply the value of $a$ by the value of $c$.
• Make sure to subtract the product of $4ac$ from the square of $b$.
• Use a calculator to simplify the expression.
• Check your work by plugging the values back into the formula.

Conclusion

In conclusion, the discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is used to determine the nature of the roots of the equation. The discriminant is an important concept in algebra because it helps us determine the nature of the roots of a quadratic equation. The discriminant has many real-world applications, including physics, engineering, and economics. By following the tips for calculating the discriminant, we can avoid common mistakes and ensure that our calculations are accurate.

Final Answer

The final answer is: $\boxed{-203}$

What is the Discriminant of a Quadratic Equation?

The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol $\Delta$ or $D$. The discriminant is used to determine the nature of the roots of the quadratic equation.

Q: What is the formula for calculating the discriminant?

A: The formula for calculating the discriminant is $\Delta = b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What does the discriminant tell us about the roots of the quadratic equation?

A: The discriminant tells us about the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: How do I calculate the discriminant if the quadratic equation is in the form $ax^2 + bx + c = 0$?

A: To calculate the discriminant, simply plug the values of $a$, $b$, and $c$ into the formula $\Delta = b^2 - 4ac$.

Q: What if the discriminant is negative? What does that mean?

A: If the discriminant is negative, it means that the quadratic equation has no real roots. This means that the equation has complex roots.

Q: Can I use the discriminant to find the roots of the quadratic equation?

A: Yes, you can use the discriminant to find the roots of the quadratic equation. If the discriminant is positive, you can use the quadratic formula to find the roots of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to find the roots of a quadratic equation. It is given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula to find the roots of the quadratic equation?

A: To use the quadratic formula, simply plug the values of $a$, $b$, and $c$ into the formula, and then simplify the expression.

Q: What if the discriminant is zero? What does that mean?

A: If the discriminant is zero, it means that the quadratic equation has one real root. This means that the equation has a repeated root.

Q: Can I use the discriminant to determine the stability of a system?

A: Yes, you can use the discriminant to determine the stability of a system. If the discriminant is positive, the system is stable. If the discriminant is negative, the system is unstable.

Q: What are some real-world applications of the discriminant?

A: The discriminant has many real-world applications, including physics, engineering, and economics. It is used to determine the stability of a system, the safety of a structure, and the competitiveness of a market.

Q: How do I avoid common mistakes when calculating the discriminant?

A: To avoid common mistakes when calculating the discriminant, make sure to square the value of $b$, multiply the value of $a$ by the value of $c$, and subtract the product of $4ac$ from the square of $b$.

Q: What are some tips for calculating the discriminant?

A: To calculate the discriminant, follow these tips:

• Make sure to square the value of $b$.
• Make sure to multiply the value of $a$ by the value of $c$.
• Make sure to subtract the product of $4ac$ from the square of $b$.
• Use a calculator to simplify the expression.
• Check your work by plugging the values back into the formula.

Conclusion

In conclusion, the discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is used to determine the nature of the roots of the quadratic equation. The discriminant has many real-world applications, including physics, engineering, and economics. By following the tips for calculating the discriminant, we can avoid common mistakes and ensure that our calculations are accurate.

Final Answer

The final answer is: $\boxed{-203}$