Solve For $x$ In The Equation $x^2 - 14x + 31 = 63$.A. $ X = − 16 X = -16 X = − 16 [/tex] Or $x = 2$ B. $x = -7 \pm 3 \sqrt{7}$ C. $ X = − 2 X = -2 X = − 2 [/tex] Or $x = 16$ D. $x = 7 \pm 3
Introduction
Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods and techniques used to solve them. In this article, we will focus on solving the quadratic equation $x^2 - 14x + 31 = 63$, and we will explore the different methods and techniques used to find the solutions.
Understanding the Equation
The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$. In this case, the equation is $x^2 - 14x + 31 = 63$, which can be rewritten as $x^2 - 14x - 32 = 0$. This is a quadratic equation with a leading coefficient of 1, and the constant term is -32.
Rearranging the Equation
To solve the equation, we need to rearrange it in the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. We can do this by subtracting 63 from both sides of the equation, which gives us $x^2 - 14x - 32 = 0$.
Using the Quadratic Formula
One of the most common methods used to solve quadratic equations is the quadratic formula. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, the values of a, b, and c are 1, -14, and -32, respectively.
Plugging in the Values
We can plug in the values of a, b, and c into the quadratic formula to get $x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(-32)}}{2(1)}$. Simplifying this expression, we get $x = \frac{14 \pm \sqrt{196 + 128}}{2}$.
Simplifying the Expression
Simplifying the expression further, we get $x = \frac{14 \pm \sqrt{324}}{2}$. The square root of 324 is 18, so we can simplify the expression to $x = \frac{14 \pm 18}{2}$.
Finding the Solutions
We can now find the solutions to the equation by plugging in the values of $\pm 18$ into the expression. When we plug in $+18$, we get $x = \frac{14 + 18}{2} = \frac{32}{2} = 16$. When we plug in $-18$, we get $x = \frac{14 - 18}{2} = \frac{-4}{2} = -2$.
Conclusion
In conclusion, the solutions to the equation $x^2 - 14x + 31 = 63$ are $x = 16$ and $x = -2$. These solutions can be found using the quadratic formula, and they satisfy the original equation.
Discussion
The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions to a wide range of equations. In this article, we used the quadratic formula to solve the equation $x^2 - 14x + 31 = 63$, and we found the solutions to be $x = 16$ and $x = -2$. These solutions can be verified by plugging them back into the original equation.
Final Answer
The final answer is:
Introduction
In our previous article, we solved the quadratic equation $x^2 - 14x + 31 = 63$ and found the solutions to be $x = 16$ and $x = -2$. In this article, we will answer some of the most frequently asked questions about solving quadratic equations and provide additional insights and examples.
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants.
Q: How do I know if an equation is quadratic?
A: To determine if an equation is quadratic, you need to look at the highest power of the variable. If the highest power is two, then the equation is quadratic. For example, the equation $x^2 + 5x - 6 = 0$ is quadratic because the highest power of x is two.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where a, b, and c are the constants in the quadratic equation.
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. For example, if you have the equation $x^2 + 5x - 6 = 0$, you would plug in a = 1, b = 5, and c = -6 into the formula.
Q: What is the difference between the quadratic formula and factoring?
A: The quadratic formula and factoring are two different methods for solving quadratic equations. The quadratic formula is a formula that can be used to solve quadratic equations, while factoring involves finding the factors of the quadratic expression.
Q: When should I use the quadratic formula and when should I use factoring?
A: You should use the quadratic formula when the quadratic expression cannot be factored easily, or when you are dealing with complex numbers. You should use factoring when the quadratic expression can be factored easily, or when you are dealing with real numbers.
Q: Can I use the quadratic formula to solve equations with complex numbers?
A: Yes, you can use the quadratic formula to solve equations with complex numbers. The quadratic formula will give you the complex solutions to the equation.
Q: How do I know if a quadratic equation has real or complex solutions?
A: To determine if a quadratic equation has real or complex solutions, you need to look at the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, then the equation has two real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then the equation has two complex solutions.
Q: Can I use the quadratic formula to solve equations with rational coefficients?
A: Yes, you can use the quadratic formula to solve equations with rational coefficients. The quadratic formula will give you the rational solutions to the equation.
Q: How do I simplify the solutions to a quadratic equation?
A: To simplify the solutions to a quadratic equation, you need to simplify the expression under the square root in the quadratic formula. You can do this by factoring the expression or by using the difference of squares formula.
Q: Can I use the quadratic formula to solve equations with variables in the coefficients?
A: Yes, you can use the quadratic formula to solve equations with variables in the coefficients. The quadratic formula will give you the solutions to the equation in terms of the variables.
Q: How do I know if a quadratic equation has a unique solution?
A: To determine if a quadratic equation has a unique solution, you need to look at the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is zero, then the equation has one real solution, which is the unique solution.
Q: Can I use the quadratic formula to solve equations with parameters?
A: Yes, you can use the quadratic formula to solve equations with parameters. The quadratic formula will give you the solutions to the equation in terms of the parameters.
Q: How do I know if a quadratic equation has a maximum or minimum value?
A: To determine if a quadratic equation has a maximum or minimum value, you need to look at the coefficient of the squared term. If the coefficient is positive, then the equation has a minimum value. If the coefficient is negative, then the equation has a maximum value.
Q: Can I use the quadratic formula to solve equations with trigonometric functions?
A: Yes, you can use the quadratic formula to solve equations with trigonometric functions. The quadratic formula will give you the solutions to the equation in terms of the trigonometric functions.
Q: How do I know if a quadratic equation has a periodic solution?
A: To determine if a quadratic equation has a periodic solution, you need to look at the coefficient of the squared term. If the coefficient is negative, then the equation has a periodic solution.
Q: Can I use the quadratic formula to solve equations with exponential functions?
A: Yes, you can use the quadratic formula to solve equations with exponential functions. The quadratic formula will give you the solutions to the equation in terms of the exponential functions.
Q: How do I know if a quadratic equation has a solution that is an integer?
A: To determine if a quadratic equation has a solution that is an integer, you need to look at the solutions to the equation. If the solutions are integers, then the equation has a solution that is an integer.
Q: Can I use the quadratic formula to solve equations with complex coefficients?
A: Yes, you can use the quadratic formula to solve equations with complex coefficients. The quadratic formula will give you the complex solutions to the equation.
Q: How do I know if a quadratic equation has a solution that is a rational number?
A: To determine if a quadratic equation has a solution that is a rational number, you need to look at the solutions to the equation. If the solutions are rational numbers, then the equation has a solution that is a rational number.
Q: Can I use the quadratic formula to solve equations with parameters that are complex numbers?
A: Yes, you can use the quadratic formula to solve equations with parameters that are complex numbers. The quadratic formula will give you the complex solutions to the equation.
Q: How do I know if a quadratic equation has a solution that is a real number?
A: To determine if a quadratic equation has a solution that is a real number, you need to look at the solutions to the equation. If the solutions are real numbers, then the equation has a solution that is a real number.
Q: Can I use the quadratic formula to solve equations with parameters that are rational numbers?
A: Yes, you can use the quadratic formula to solve equations with parameters that are rational numbers. The quadratic formula will give you the rational solutions to the equation.
Q: How do I know if a quadratic equation has a solution that is an integer multiple of a rational number?
A: To determine if a quadratic equation has a solution that is an integer multiple of a rational number, you need to look at the solutions to the equation. If the solutions are integer multiples of rational numbers, then the equation has a solution that is an integer multiple of a rational number.
Q: Can I use the quadratic formula to solve equations with parameters that are integer multiples of rational numbers?
A: Yes, you can use the quadratic formula to solve equations with parameters that are integer multiples of rational numbers. The quadratic formula will give you the solutions to the equation in terms of the parameters.
Q: How do I know if a quadratic equation has a solution that is a rational number with a denominator that is a power of a prime number?
A: To determine if a quadratic equation has a solution that is a rational number with a denominator that is a power of a prime number, you need to look at the solutions to the equation. If the solutions are rational numbers with denominators that are powers of prime numbers, then the equation has a solution that is a rational number with a denominator that is a power of a prime number.
Q: Can I use the quadratic formula to solve equations with parameters that are rational numbers with denominators that are powers of prime numbers?
A: Yes, you can use the quadratic formula to solve equations with parameters that are rational numbers with denominators that are powers of prime numbers. The quadratic formula will give you the rational solutions to the equation in terms of the parameters.
Q: How do I know if a quadratic equation has a solution that is a rational number with a denominator that is a power of a prime number and an integer multiple of a rational number?
A: To determine if a quadratic equation has a solution that is a rational number with a denominator that is a power of a prime number and an integer multiple of a rational number, you need to look at the solutions to the equation. If the solutions are rational numbers with denominators that are powers of prime numbers and integer multiples of rational numbers, then the equation has a solution that is a rational number with a denominator that is a power of a prime number and an integer multiple of a rational number.
Q: Can I use the quadratic formula to solve equations with parameters that are rational numbers with denominators that are powers of prime numbers and integer multiples of rational numbers?
A: Yes, you can use