For The Following Quadratic Equation, Find The Discriminant.$\[5x^2 - 16x - 253 = 4x + 7\\]Answer Attempt 1 Out Of 2.

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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. It has the general form of ax2+bx+c=0{ax^2 + bx + c = 0}, where a{a}, b{b}, and c{c} are constants, and x{x} is the variable. In this case, we are given the quadratic equation 5x2−16x−253=4x+7{5x^2 - 16x - 253 = 4x + 7}. To find the discriminant, we first need to rewrite the equation in the standard form of a quadratic equation.

Rewriting the Quadratic Equation

To rewrite the equation in the standard form, we need to move all the terms to one side of the equation. We can do this by subtracting 4x{4x} from both sides and adding 7{7} to both sides. This gives us:

5x2−16x−253−4x−7=0{5x^2 - 16x - 253 - 4x - 7 = 0}

Simplifying the equation, we get:

5x2−20x−260=0{5x^2 - 20x - 260 = 0}

The Discriminant Formula

The discriminant of a quadratic equation is given by the formula b2−4ac{b^2 - 4ac}, where a{a}, b{b}, and c{c} are the coefficients of the quadratic equation. In this case, a=5{a = 5}, b=−20{b = -20}, and c=−260{c = -260}. Plugging these values into the formula, we get:

(−20)2−4(5)(−260){(-20)^2 - 4(5)(-260)}

Simplifying the Expression

To simplify the expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the exponent: (−20)2=400{(-20)^2 = 400}
  2. Multiply 4{4} and 5{5}: 4(5)=20{4(5) = 20}
  3. Multiply 20{20} and −260{-260}: 20(−260)=−5200{20(-260) = -5200}
  4. Subtract 5200{5200} from 400{400}: 400−5200=−4800{400 - 5200 = -4800}

The Final Answer

Therefore, the discriminant of the quadratic equation 5x2−16x−253=4x+7{5x^2 - 16x - 253 = 4x + 7} is −4800{-4800}.

Conclusion

In this article, we have learned how to find the discriminant of a quadratic equation. We started by rewriting the given equation in the standard form, and then we used the formula b2−4ac{b^2 - 4ac} to find the discriminant. We simplified the expression by following the order of operations, and finally, we arrived at the final answer. The discriminant is an important concept in algebra, and it has many applications in various fields, including physics, engineering, and economics.

Real-World Applications

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 design stable structures, such as bridges and buildings. In economics, the discriminant is used to analyze the behavior of economic systems.

Tips and Tricks

Here are some tips and tricks to help you find the discriminant of a quadratic equation:

  • Make sure to rewrite the equation in the standard form before finding the discriminant.
  • Use the formula b2−4ac{b^2 - 4ac} to find the discriminant.
  • Simplify the expression by following the order of operations.
  • Check your answer by plugging it back into the original equation.

Practice Problems

Here are some practice problems to help you practice finding the discriminant of a quadratic equation:

  • Find the discriminant of the quadratic equation x2+5x+6=0{x^2 + 5x + 6 = 0}.
  • Find the discriminant of the quadratic equation 2x2−3x−1=0{2x^2 - 3x - 1 = 0}.
  • Find the discriminant of the quadratic equation x2−4x+4=0{x^2 - 4x + 4 = 0}.

Conclusion

In conclusion, finding the discriminant of a quadratic equation is an important concept in algebra. We have learned how to rewrite the equation in the standard form, use the formula b2−4ac{b^2 - 4ac} to find the discriminant, and simplify the expression by following the order of operations. We have also seen some real-world applications of the discriminant and some tips and tricks to help you find the discriminant. With practice, you will become proficient in finding the discriminant of a quadratic equation.

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the quadratic equation discriminant.

Q: What is the discriminant of a quadratic equation?

A: 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.

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

A: To find the discriminant of a quadratic equation, you need to use the formula b2−4ac{b^2 - 4ac}, where a{a}, b{b}, and c{c} are the coefficients of the equation.

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

A: The discriminant tells you whether the roots of the equation are real or complex. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

Q: How do I determine the nature of the roots of the equation?

A: To determine the nature of the roots of the equation, you need to calculate the discriminant and then use the following rules:

  • If the discriminant is positive, the roots are real and distinct.
  • If the discriminant is zero, the roots are real and equal.
  • If the discriminant is negative, the roots are complex.

Q: Can I use the discriminant to solve quadratic equations?

A: Yes, you can use the discriminant to solve quadratic equations. If the discriminant is positive, you can use the quadratic formula to find the roots of the equation. If the discriminant is zero, you can use the quadratic formula to find the repeated root of the equation. If the discriminant is negative, you can use the quadratic formula to find the complex roots of the equation.

Q: What are some common mistakes to avoid when finding the discriminant?

A: Some common mistakes to avoid when finding the discriminant include:

  • Not rewriting the equation in the standard form before finding the discriminant.
  • Not using the correct formula to find the discriminant.
  • Not simplifying the expression correctly.
  • Not checking the answer by plugging it back into the original equation.

Q: How can I practice finding the discriminant of quadratic equations?

A: You can practice finding the discriminant of quadratic equations by working through examples and exercises. You can also use online resources and calculators to help you practice.

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

A: The discriminant has many real-world applications, including:

  • Determining the stability of a system in physics.
  • Designing stable structures in engineering.
  • Analyzing the behavior of economic systems in economics.

Q: Can I use the discriminant to solve systems of equations?

A: Yes, you can use the discriminant to solve systems of equations. However, this is a more advanced topic and requires a good understanding of linear algebra and matrix theory.

Q: What are some tips and tricks for finding the discriminant?

A: Some tips and tricks for finding the discriminant include:

  • Make sure to rewrite the equation in the standard form before finding the discriminant.
  • Use the formula b2−4ac{b^2 - 4ac} to find the discriminant.
  • Simplify the expression correctly.
  • Check the answer by plugging it back into the original equation.

Conclusion

In conclusion, the discriminant is an important concept in algebra that can be used to determine the nature of the roots of a quadratic equation. We have answered some frequently asked questions about the discriminant and provided some tips and tricks for finding it. With practice, you will become proficient in finding the discriminant of quadratic equations.