Enter The Correct Answer In The Box.What Are The Solutions Of This Quadratic Equation? X 2 = 16 X − 65 X^2 = 16x - 65 X 2 = 16 X − 65 Substitute The Values Of A A A And B B B To Complete The Solutions.

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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 explore the solutions to the quadratic equation x2=16x−65x^2 = 16x - 65. We will break down the steps involved in solving this equation and provide a clear understanding of the process.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, xx) is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Equation x2=16x−65x^2 = 16x - 65

The given quadratic equation is x2=16x−65x^2 = 16x - 65. To solve this equation, we need to rewrite it in the standard form ax2+bx+c=0ax^2 + bx + c = 0. We can do this by subtracting 16x16x from both sides of the equation and adding 6565 to both sides:

x2−16x+65=0x^2 - 16x + 65 = 0

Now, we have a quadratic equation in the standard form. We can see that the coefficient of x2x^2 is 11, which means a=1a = 1. The coefficient of xx is −16-16, which means b=−16b = -16. The constant term is 6565, which means c=65c = 65.

Completing the Square

To solve the quadratic equation, we can use the method of completing the square. This method involves rewriting the quadratic equation in a form that allows us to easily find the solutions. We can start by dividing both sides of the equation by the coefficient of x2x^2, which is 11:

x2−16x=−65x^2 - 16x = -65

Next, we can add the square of half the coefficient of xx to both sides of the equation. In this case, half of −16-16 is −8-8, and the square of −8-8 is 6464. We can add 6464 to both sides of the equation:

x2−16x+64=−65+64x^2 - 16x + 64 = -65 + 64

Simplifying the right-hand side of the equation, we get:

x2−16x+64=−1x^2 - 16x + 64 = -1

Now, we can rewrite the left-hand side of the equation as a perfect square:

(x−8)2=−1(x - 8)^2 = -1

Solving for xx

To solve for xx, we can take the square root of both sides of the equation. Since the square root of −1-1 is an imaginary number, we can write:

x−8=±−1x - 8 = \pm \sqrt{-1}

Simplifying the right-hand side of the equation, we get:

x−8=±ix - 8 = \pm i

where ii is the imaginary unit, which satisfies the equation i2=−1i^2 = -1. Adding 88 to both sides of the equation, we get:

x=8±ix = 8 \pm i

Conclusion

In this article, we solved the quadratic equation x2=16x−65x^2 = 16x - 65 using the method of completing the square. We rewrote the equation in a form that allowed us to easily find the solutions, and we used the quadratic formula to solve for xx. The solutions to the equation are x=8+ix = 8 + i and x=8−ix = 8 - i. We hope this article has provided a clear understanding of the process involved in solving quadratic equations.

Final Answer

The final answer is: 8+i,8−i\boxed{8 + i, 8 - i}

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important topic.

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 (in this case, xx) is two. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The method you choose will depend on the specific equation and the type of solutions you are looking for.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the coefficients aa, bb, and cc in the quadratic equation. Then, you can plug these values into the formula and simplify to find the solutions.

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are the values of xx that satisfy the equation. These solutions can be real or complex numbers, depending on the specific equation.

Q: How do I determine the number of solutions to a quadratic equation?

A: The number of solutions to a quadratic equation depends on the discriminant, which is given by b2−4acb^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 two complex solutions.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared term, while a linear equation does not.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring if it can be written in the form (x−r)(x−s)=0(x - r)(x - s) = 0, where rr and ss are constants.

Q: What is the significance of the quadratic equation in real-life applications?

A: The quadratic equation has numerous applications in real-life situations, such as physics, engineering, economics, and computer science. It is used to model and solve problems involving quadratic relationships.

Conclusion

In this article, we have addressed some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important topic. We hope this article has been helpful in clarifying any doubts you may have had about quadratic equations.

Final Answer

The final answer is: Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals.