Write The Quadratic Equation In Standard Form:${ 4x^2 - 5 = 7x }$
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Introduction
In mathematics, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. In this article, we will discuss how to write the quadratic equation in standard form, using the given equation 4x^2 - 5 = 7x as an example.
Understanding the Quadratic Equation
A quadratic equation can be written in the form of ax^2 + bx + c = 0, where a, b, and c are constants. The equation can be solved using various methods, including factoring, completing the square, and the quadratic formula. In this article, we will focus on writing the quadratic equation in standard form.
Writing the Quadratic Equation in Standard Form
To write the quadratic equation in standard form, we need to isolate the x^2 term on one side of the equation. We can do this by subtracting 7x from both sides of the equation and adding 5 to both sides.
Step 1: Subtract 7x from both sides
Subtracting 7x from both sides of the equation gives us:
4x^2 - 5 - 7x = 0
Step 2: Add 5 to both sides
Adding 5 to both sides of the equation gives us:
4x^2 - 7x = 5
Step 3: Rearrange the equation
Rearranging the equation to put the x^2 term first gives us:
4x^2 - 7x - 5 = 0
Conclusion
In this article, we discussed how to write the quadratic equation in standard form using the given equation 4x^2 - 5 = 7x as an example. We followed the steps of subtracting 7x from both sides, adding 5 to both sides, and rearranging the equation to put the x^2 term first. The resulting equation is 4x^2 - 7x - 5 = 0, which is in the standard form of a quadratic equation.
Example Problems
Problem 1
Write the quadratic equation in standard form: 2x^2 + 3 = 5x
Solution
Subtracting 5x from both sides gives us:
2x^2 + 3 - 5x = 0
Adding 5x to both sides gives us:
2x^2 - 5x = -3
Rearranging the equation gives us:
2x^2 - 5x + 3 = 0
Problem 2
Write the quadratic equation in standard form: x^2 - 2 = 3x
Solution
Subtracting 3x from both sides gives us:
x^2 - 3x = 2
Rearranging the equation gives us:
x^2 - 3x - 2 = 0
Tips and Tricks
- When writing a quadratic equation in standard form, make sure to isolate the x^2 term on one side of the equation.
- Use the steps of subtracting and adding to rearrange the equation and put the x^2 term first.
- Check your work by plugging the equation back into the original equation to make sure it is true.
Conclusion
In conclusion, writing a quadratic equation in standard form is an important skill in mathematics. By following the steps of subtracting and adding, and rearranging the equation, we can write a quadratic equation in standard form. Remember to check your work by plugging the equation back into the original equation to make sure it is true. With practice and patience, you will become proficient in writing quadratic equations in standard form.
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Introduction
In our previous article, we discussed how to write a quadratic equation in standard form. In this article, we will answer some frequently asked questions about quadratic equations in standard form.
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 is two. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.
Q: How do I write a quadratic equation in standard form?
A: To write a quadratic equation in standard form, you need to isolate the x^2 term on one side of the equation. You can do this by subtracting and adding to rearrange the equation and put the x^2 term first.
Q: What are the steps to write a quadratic equation in standard form?
A: The steps to write a quadratic equation in standard form are:
- Subtract the term with the variable from both sides of the equation.
- Add the constant term to both sides of the equation.
- Rearrange the equation to put the x^2 term first.
Q: How do I check my work when writing a quadratic equation in standard form?
A: To check your work, plug the equation back into the original equation to make sure it is true. If the equation is true, then you have written the quadratic equation in standard form correctly.
Q: What are some common mistakes to avoid when writing a quadratic equation in standard form?
A: Some common mistakes to avoid when writing a quadratic equation in standard form include:
- Not isolating the x^2 term on one side of the equation.
- Not adding and subtracting correctly.
- Not rearranging the equation to put the x^2 term first.
Q: Can I use a calculator to write a quadratic equation in standard form?
A: Yes, you can use a calculator to write a quadratic equation in standard form. However, it is still important to understand the steps and process of writing a quadratic equation in standard form.
Q: How do I solve a quadratic equation in standard form?
A: To solve a quadratic equation in standard form, you can use various methods, including factoring, completing the square, and the quadratic formula.
Q: What are some real-world applications of quadratic equations in standard form?
A: Quadratic equations in standard form have many real-world applications, including:
- Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity.
- Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
- Economics: Quadratic equations are used to model and analyze economic systems.
Conclusion
In conclusion, writing a quadratic equation in standard form is an important skill in mathematics. By understanding the steps and process of writing a quadratic equation in standard form, you can solve quadratic equations and apply them to real-world problems. Remember to check your work and avoid common mistakes when writing a quadratic equation in standard form.
Example Problems
Problem 1
Write the quadratic equation in standard form: 2x^2 + 3 = 5x
Solution
Subtracting 5x from both sides gives us:
2x^2 + 3 - 5x = 0
Adding 5x to both sides gives us:
2x^2 - 5x = -3
Rearranging the equation gives us:
2x^2 - 5x + 3 = 0
Problem 2
Write the quadratic equation in standard form: x^2 - 2 = 3x
Solution
Subtracting 3x from both sides gives us:
x^2 - 3x = 2
Rearranging the equation gives us:
x^2 - 3x - 2 = 0
Tips and Tricks
- When writing a quadratic equation in standard form, make sure to isolate the x^2 term on one side of the equation.
- Use the steps of subtracting and adding to rearrange the equation and put the x^2 term first.
- Check your work by plugging the equation back into the original equation to make sure it is true.
Conclusion
In conclusion, writing a quadratic equation in standard form is an important skill in mathematics. By understanding the steps and process of writing a quadratic equation in standard form, you can solve quadratic equations and apply them to real-world problems. Remember to check your work and avoid common mistakes when writing a quadratic equation in standard form.