Algebra II: Quadratics and shifts | Quadratic equations | Algebra I | Khan Academy

TL;DR

Simplify equation by multiplying both sides by x squared, use quadratic equation to solve, find multiple solutions. Shift graphs up and to the right using addition and subtraction.

Transcript

We're on problem 32. What are the solutions to the equation 1 plus 1 over x squared is equal to 3 over x? So at first. This looks like a pretty daunting equation. You have these x's in the denominator and x squared in the denominator. But I think we can simplify it if we can just get rid of these x squares in the denominator. The easiest way to do ... Read More

Key Insights

• 🍉 Equations with complex terms can be simplified by multiplying both sides to eliminate terms in the denominator.
• 🔨 The quadratic equation is a useful tool for solving complex quadratic equations when factoring is not possible.
• ❣️ Graph transformations can be achieved by adding or subtracting constants in the equation, resulting in shifts in the y and x directions.
• 😀 Shifting a graph in the y direction can be easily determined by the positive or negative value of the constant added or subtracted.

Questions & Answers

Q: How can the equation 1 + 1/x^2 = 3/x be simplified and solved?

To simplify the equation, you can multiply both sides by x^2 to eliminate the x terms in the denominator. This results in x^2 - 3x + 1 = 0. To solve for x, you can use the quadratic equation by substituting the values of a, b, and c, which yields two possible solutions: x = (3 + √5)/2 or x = (3 - √5)/2.

Q: What steps can be taken to solve the equation x^2 - 8x = 9 using the method of completing the square?

To solve the equation using completing the square, you add a number to both sides that will make the left side a perfect square. In this case, the number is 16. So, you add 16 to both sides, resulting in x^2 - 8x + 16 = 25. This can be simplified as (x - 4)^2 = 25, and you can then solve for x by taking the square root of both sides.

Q: How can you determine the shift in y for a given equation transformation?

When given an equation transformation, the shift in the y direction is determined by the constant value added or subtracted to the original equation. If the constant is positive, the graph is shifted upwards. If it is negative, the graph is shifted downwards.

Q: What is the method for determining if a graph is shifted to the left or right?

To determine if a graph is shifted to the left or right, you can analyze the values inside the parentheses of the equation. If the value is positive, the graph is shifted to the left. If it is negative, the graph is shifted to the right.

Summary & Key Takeaways

• The provided content discusses the process of simplifying and solving complex equations using various techniques.

• It also demonstrates the use of the quadratic equation when dealing with complex quadratic equations.

• Additionally, the content explains graph transformations, specifically shifting graphs up and to the right using addition and subtraction.