Write The Quadratic Equation In Standard Form:${ X^2 - 6x - 11 = -4 }$
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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. In this article, we will focus on solving the quadratic equation in standard form, which is a crucial step in understanding the behavior of quadratic functions. We will break down the solution into manageable steps, making it easy to follow and understand.
Understanding the Quadratic Equation
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. The standard 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 equation, a cannot be zero, as it would not be a quadratic equation.
The Given Equation
The given equation is:
x^2 - 6x - 11 = -4
To solve this equation, we need to isolate the variable x. The first step is to add 4 to both sides of the equation to get rid of the negative term on the right-hand side.
Adding 4 to Both Sides
x^2 - 6x - 11 + 4 = -4 + 4
This simplifies to:
x^2 - 6x - 7 = 0
Rearranging the Equation
Now that we have the equation in the standard form, we can see that it is a quadratic equation. The next step is to factor the left-hand side of the equation, if possible.
Factoring the Equation
Unfortunately, this equation does not factor easily, so we will need to use other methods to solve it. One method is to use the quadratic formula, which is:
x = (-b Β± β(b^2 - 4ac)) / 2a
In this case, a = 1, b = -6, and c = -7.
Applying the Quadratic Formula
Substituting the values of a, b, and c into the quadratic formula, we get:
x = (6 Β± β((-6)^2 - 4(1)(-7))) / 2(1)
Simplifying the expression under the square root, we get:
x = (6 Β± β(36 + 28)) / 2
x = (6 Β± β64) / 2
x = (6 Β± 8) / 2
Solving for x
Now that we have the expression under the square root simplified, we can solve for x. We have two possible solutions:
x = (6 + 8) / 2
x = 14 / 2
x = 7
x = (6 - 8) / 2
x = -2 / 2
x = -1
Conclusion
In this article, we solved the quadratic equation in standard form using the quadratic formula. We started with the given equation, added 4 to both sides to get rid of the negative term, and then rearranged the equation to get it in the standard form. We then factored the left-hand side, but since it did not factor easily, we used the quadratic formula to solve for x. We obtained two possible solutions, x = 7 and x = -1.
Final Answer
The final answer is x = 7 or x = -1.
Additional Resources
For more information on quadratic equations and how to solve them, check out the following resources:
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Equation Solver
- Wolfram Alpha: Quadratic Equation Solver
FAQs
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 (usually x) is two.
Q: How do I solve a quadratic equation?
A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and graphing.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that can be used to solve quadratic equations. It is:
x = (-b Β± β(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula?
A: To use the quadratic formula, substitute the values of a, b, and c into the formula, and then simplify the expression under the square root.
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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. In this article, we will answer some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you understand and solve these equations.
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 (usually x) is two. The standard form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and x is the variable.
Q: How do I solve a quadratic equation?
A: There are several methods to solve a quadratic equation, including:
- Factoring: If the equation can be factored easily, you can solve it by finding the factors.
- Using the quadratic formula: The quadratic formula is a formula that can be used to solve quadratic equations. It is:
x = (-b Β± β(b^2 - 4ac)) / 2a
- Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that can be used to solve quadratic equations. It is:
x = (-b Β± β(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula?
A: To use the quadratic formula, substitute the values of a, b, and c into the formula, and then simplify the expression under the square root.
Q: What is the discriminant?
A: The discriminant is the expression under the square root in the quadratic formula, which is:
b^2 - 4ac
If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
Q: How do I determine the number of solutions to a quadratic equation?
A: To determine the number of solutions to a quadratic equation, you can use the discriminant. If the discriminant is:
- Positive, the equation has two distinct real solutions.
- Zero, the equation has one real solution.
- Negative, the equation has no real solutions.
Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?
A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. If the discriminant is negative, the quadratic formula will give you two complex solutions.
Q: How do I simplify the expression under the square root in the quadratic formula?
A: To simplify the expression under the square root, you can use the following steps:
- Factor the expression under the square root, if possible.
- Simplify the expression under the square root by combining like terms.
- Take the square root of the simplified expression.
Q: Can I use the quadratic formula to solve a quadratic equation with a variable coefficient?
A: Yes, you can use the quadratic formula to solve a quadratic equation with a variable coefficient. However, you will need to substitute the values of the variable coefficients into the formula.
Q: How do I check my solutions to a quadratic equation?
A: To check your solutions to a quadratic equation, you can substitute the solutions back into the original equation and verify that they are true.
Conclusion
In this article, we have answered some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you understand and solve these equations. Whether you are a student or a professional, this article will help you to master the art of solving quadratic equations.
Additional Resources
For more information on quadratic equations and how to solve them, check out the following resources:
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Equation Solver
- Wolfram Alpha: Quadratic Equation Solver
FAQs
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 (usually x) is two.
Q: How do I solve a quadratic equation?
A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and graphing.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that can be used to solve quadratic equations. It is:
x = (-b Β± β(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula?
A: To use the quadratic formula, substitute the values of a, b, and c into the formula, and then simplify the expression under the square root.
Q: What is the discriminant?
A: The discriminant is the expression under the square root in the quadratic formula, which is:
b^2 - 4ac
If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.