Solve Each Equation Using The Quadratic Formula. Round To The Nearest Hundredth.19. \[$-2x^2 + 12x - 5 = 0\$\]20. \[$x^2 + 19x - 7 = 0\$\]
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to solve quadratic equations using the quadratic formula. We will also provide step-by-step solutions to two quadratic equations, rounding our answers to the nearest hundredth.
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 given by:
x = (-b ± √(b^2 - 4ac)) / 2a
How to Use the Quadratic Formula
To use the quadratic formula, we need to identify the values of a, b, and c in the quadratic equation. We then plug these values into the formula and simplify to find the solutions.
Step 1: Identify the Values of a, b, and c
In the quadratic equation ax^2 + bx + c = 0, the values of a, b, and c are:
- a: the coefficient of the x^2 term
- b: the coefficient of the x term
- c: the constant term
Step 2: Plug the Values into the Quadratic Formula
Once we have identified the values of a, b, and c, we plug them into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Step 3: Simplify the Expression
We then simplify the expression by evaluating the square root and dividing by 2a.
Solving the First Quadratic Equation
Let's solve the quadratic equation -2x^2 + 12x - 5 = 0 using the quadratic formula.
Step 1: Identify the Values of a, b, and c
In the quadratic equation -2x^2 + 12x - 5 = 0, the values of a, b, and c are:
- a: -2
- b: 12
- c: -5
Step 2: Plug the Values into the Quadratic Formula
We plug the values of a, b, and c into the quadratic formula:
x = (-(12) ± √((12)^2 - 4(-2)(-5))) / 2(-2)
Step 3: Simplify the Expression
We simplify the expression by evaluating the square root and dividing by 2a:
x = (-12 ± √(144 - 40)) / -4 x = (-12 ± √104) / -4 x = (-12 ± 10.2) / -4
Solutions
We have two possible solutions:
x = (-12 + 10.2) / -4 x = -1.8 / -4 x = 0.45
x = (-12 - 10.2) / -4 x = -22.2 / -4 x = 5.55
Solving the Second Quadratic Equation
Let's solve the quadratic equation x^2 + 19x - 7 = 0 using the quadratic formula.
Step 1: Identify the Values of a, b, and c
In the quadratic equation x^2 + 19x - 7 = 0, the values of a, b, and c are:
- a: 1
- b: 19
- c: -7
Step 2: Plug the Values into the Quadratic Formula
We plug the values of a, b, and c into the quadratic formula:
x = -(19) ± √((19)^2 - 4(1)(-7)) / 2(1)
Step 3: Simplify the Expression
We simplify the expression by evaluating the square root and dividing by 2a:
x = (-19 ± √(361 + 28)) / 2 x = (-19 ± √389) / 2 x = (-19 ± 19.7) / 2
Solutions
We have two possible solutions:
x = (-19 + 19.7) / 2 x = 0.7 / 2 x = 0.35
x = (-19 - 19.7) / 2 x = -38.7 / 2 x = -19.35
Conclusion
Introduction
In our previous article, we explored how to solve quadratic equations using the quadratic formula. In this article, we will answer some frequently asked questions about the quadratic formula and provide additional examples to help you understand this powerful mathematical tool.
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 given by:
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. You then plug these values into the formula and simplify to find the solutions.
Q: What are the values of a, b, and c?
A: In the quadratic equation ax^2 + bx + c = 0, the values of a, b, and c are:
- a: the coefficient of the x^2 term
- b: the coefficient of the x term
- c: the constant term
Q: How do I simplify the expression?
A: To simplify the expression, you need to evaluate the square root and divide by 2a.
Q: What if the discriminant is negative?
A: If the discriminant (b^2 - 4ac) is negative, then the quadratic equation has no real solutions. In this case, the quadratic formula will give you complex solutions.
Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?
A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you need to be careful when simplifying the expression, as complex numbers can be tricky to work with.
Q: Are there any other ways to solve quadratic equations?
A: Yes, there are other ways to solve quadratic equations, such as factoring, completing the square, and using the quadratic formula with complex coefficients.
Q: Can I use the quadratic formula to solve quadratic equations with fractional coefficients?
A: Yes, you can use the quadratic formula to solve quadratic equations with fractional coefficients. However, you need to be careful when simplifying the expression, as fractional numbers can be tricky to work with.
Q: How do I round my answers to the nearest hundredth?
A: To round your answers to the nearest hundredth, you need to look at the thousandths place and decide whether to round up or down. If the thousandths place is 5 or greater, you round up. If the thousandths place is less than 5, you round down.
Examples
Let's solve some quadratic equations using the quadratic formula.
Example 1
Solve the quadratic equation x^2 + 5x + 6 = 0 using the quadratic formula.
Step 1: Identify the values of a, b, and c
In the quadratic equation x^2 + 5x + 6 = 0, the values of a, b, and c are:
- a: 1
- b: 5
- c: 6
Step 2: Plug the values into the quadratic formula
We plug the values of a, b, and c into the quadratic formula:
x = -(5) ± √((5)^2 - 4(1)(6)) / 2(1)
Step 3: Simplify the expression
We simplify the expression by evaluating the square root and dividing by 2a:
x = (-5 ± √(25 - 24)) / 2 x = (-5 ± √1) / 2 x = (-5 ± 1) / 2
Solutions
We have two possible solutions:
x = (-5 + 1) / 2 x = -4 / 2 x = -2
x = (-5 - 1) / 2 x = -6 / 2 x = -3
Example 2
Solve the quadratic equation x^2 - 4x + 4 = 0 using the quadratic formula.
Step 1: Identify the values of a, b, and c
In the quadratic equation x^2 - 4x + 4 = 0, the values of a, b, and c are:
- a: 1
- b: -4
- c: 4
Step 2: Plug the values into the quadratic formula
We plug the values of a, b, and c into the quadratic formula:
x = -(-4) ± √((-4)^2 - 4(1)(4)) / 2(1)
Step 3: Simplify the expression
We simplify the expression by evaluating the square root and dividing by 2a:
x = 4 ± √(16 - 16) / 2 x = 4 ± √0 / 2 x = 4 / 2 x = 2
Solution
We have one possible solution:
x = 2
Conclusion
In this article, we have answered some frequently asked questions about the quadratic formula and provided additional examples to help you understand this powerful mathematical tool. We have also explored how to use the quadratic formula to solve quadratic equations with complex coefficients and fractional coefficients. With practice and patience, you will become proficient in using the quadratic formula to solve a wide range of mathematical problems.