Solve Each Equation Using The Quadratic Formula. Round To The Nearest Hundredth.19. $-2x^2 + 12x - 5 = 0$20. $x^2 + 19x - 7 = 0$21. $3x^2 + 18x - 27 = 0$22. $-7x^2 + 2x + 1 = 0$23. $2x^2 + 9x + 7 = 0$24.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and in this article, we will explore how to use it to solve a variety of quadratic equations. We will also discuss the importance of rounding to the nearest hundredth and provide step-by-step solutions to each equation.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is as follows:

x = (-b ± √(b^2 - 4ac)) / 2a

How to Use the Quadratic Formula

To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, you can plug these values into the formula and simplify to find the solutions.

Solving Quadratic Equations

19. −2x2+12x−5=0-2x^2 + 12x - 5 = 0

To solve this equation, we need to identify the values of a, b, and c. In this case, a = -2, b = 12, and c = -5.

x = (-b ± √(b^2 - 4ac)) / 2a x = (-(12) ± √((12)^2 - 4(-2)(-5))) / 2(-2) x = (-12 ± √(144 - 40)) / -4 x = (-12 ± √104) / -4 x = (-12 ± 10.2) / -4

Simplifying, we get two possible solutions:

x = (-12 + 10.2) / -4 = -1.8 / -4 = 0.45 x = (-12 - 10.2) / -4 = -22.2 / -4 = 5.55

Rounding to the nearest hundredth, we get:

x ≈ 0.45 x ≈ 5.55

20. x2+19x−7=0x^2 + 19x - 7 = 0

To solve this equation, we need to identify the values of a, b, and c. In this case, a = 1, b = 19, and c = -7.

x = (-b ± √(b^2 - 4ac)) / 2a x = (-(19) ± √((19)^2 - 4(1)(-7))) / 2(1) x = (-19 ± √(361 + 28)) / 2 x = (-19 ± √389) / 2

Simplifying, we get two possible solutions:

x = (-19 + √389) / 2 x = (-19 - √389) / 2

Using a calculator to approximate the value of √389, we get:

x ≈ (-19 + 19.7) / 2 = 0.35 x ≈ (-19 - 19.7) / 2 = -19.35

Rounding to the nearest hundredth, we get:

x ≈ 0.35 x ≈ -19.35

21. 3x2+18x−27=03x^2 + 18x - 27 = 0

To solve this equation, we need to identify the values of a, b, and c. In this case, a = 3, b = 18, and c = -27.

x = (-b ± √(b^2 - 4ac)) / 2a x = (-(18) ± √((18)^2 - 4(3)(-27))) / 2(3) x = (-18 ± √(324 + 324)) / 6 x = (-18 ± √648) / 6

Simplifying, we get two possible solutions:

x = (-18 + √648) / 6 x = (-18 - √648) / 6

Using a calculator to approximate the value of √648, we get:

x ≈ (-18 + 25.46) / 6 = 2.08 x ≈ (-18 - 25.46) / 6 = -5.08

Rounding to the nearest hundredth, we get:

x ≈ 2.08 x ≈ -5.08

22. −7x2+2x+1=0-7x^2 + 2x + 1 = 0

To solve this equation, we need to identify the values of a, b, and c. In this case, a = -7, b = 2, and c = 1.

x = (-b ± √(b^2 - 4ac)) / 2a x = (-(2) ± √((2)^2 - 4(-7)(1))) / 2(-7) x = (-2 ± √(4 + 28)) / -14 x = (-2 ± √32) / -14

Simplifying, we get two possible solutions:

x = (-2 + √32) / -14 x = (-2 - √32) / -14

Using a calculator to approximate the value of √32, we get:

x ≈ (-2 + 5.66) / -14 = -0.39 x ≈ (-2 - 5.66) / -14 = 0.39

Rounding to the nearest hundredth, we get:

x ≈ -0.39 x ≈ 0.39

23. 2x2+9x+7=02x^2 + 9x + 7 = 0

To solve this equation, we need to identify the values of a, b, and c. In this case, a = 2, b = 9, and c = 7.

x = (-b ± √(b^2 - 4ac)) / 2a x = (-(9) ± √((9)^2 - 4(2)(7))) / 2(2) x = (-9 ± √(81 - 56)) / 4 x = (-9 ± √25) / 4

Simplifying, we get two possible solutions:

x = (-9 + √25) / 4 x = (-9 - √25) / 4

Using a calculator to approximate the value of √25, we get:

x ≈ (-9 + 5) / 4 = -1 x ≈ (-9 - 5) / 4 = -3.5

Rounding to the nearest hundredth, we get:

x ≈ -1.00 x ≈ -3.50

Conclusion

Frequently Asked Questions

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is as follows:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, you can plug these values into the formula and simplify to find the solutions.

Q: What are the values of a, b, and c in the quadratic formula?

A: In the quadratic formula, a is the coefficient of the x^2 term, b is the coefficient of the x term, and c is the constant term.

Q: How do I simplify the quadratic formula?

A: To simplify the quadratic formula, you need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the square root (b^2 - 4ac)
  2. Simplify the square root
  3. Simplify the fraction (x = (-b ± √(b^2 - 4ac)) / 2a)

Q: What if the expression inside the square root is negative?

A: If the expression inside the square root is negative, then the quadratic equation has no real solutions. In this case, the solutions will be complex numbers.

Q: How do I round the solutions to the nearest hundredth?

A: To round the solutions to the nearest hundredth, you need to look at the thousandths place. If the digit in the thousandths place is 5 or greater, you need to round up. If the digit in the thousandths place is less than 5, you need to round down.

Q: Can I use the quadratic formula to solve quadratic equations with complex solutions?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex solutions. However, you need to be careful when simplifying the square root and the fraction.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

  • Not identifying the values of a, b, and c correctly
  • Not simplifying the expression inside the square root correctly
  • Not simplifying the square root correctly
  • Not simplifying the fraction correctly
  • Not rounding the solutions to the nearest hundredth correctly

Q: Can I use the quadratic formula to solve quadratic equations with rational solutions?

A: Yes, you can use the quadratic formula to solve quadratic equations with rational solutions. However, you need to be careful when simplifying the square root and the fraction.

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

A: The quadratic formula has many real-world applications, including:

  • Physics: The quadratic formula is used to solve problems involving motion, such as the trajectory of a projectile.
  • Engineering: The quadratic formula is used to solve problems involving stress and strain on materials.
  • Computer Science: The quadratic formula is used to solve problems involving algorithms and data structures.

Conclusion

In this article, we have answered some frequently asked questions about the quadratic formula. We have discussed the basics of the quadratic formula, how to use it, and some common mistakes to avoid. We have also discussed some real-world applications of the quadratic formula. By following these tips and using the quadratic formula, you can solve quadratic equations with ease and accuracy.