find the roots of x²-10x=17 using completing the square​

Answer:The two possible answers are: x = 5 + (42)^(1/2) x = 5 - (42)^(1/2)Step-by-step explanation:Completing the square is a method used in algebra to rewrite a quadratic expression or equation in a form that makes it easier to solve or analyze. The main goal is to transform a quadratic expression of the form ax^2+bx+c into a perfect square trinomial plus or minus a constant. This technique is especially useful for solving quadratic equations, graphing parabolas, and deriving the quadratic formula.What Does It Mean to "Complete the Square"?To "complete the square" means to add and subtract a specific value to a quadratic expression so that part of the expression becomes a perfect square trinomial. A perfect square trinomial is an expression that can be written as (x+d)^2, where d is a constant. This process allows us to rewrite the quadratic in a way that makes it easier to solve for x or to see important features of its graph, such as the vertex.Step-by-Step Guide to Completing the SquareLet's break down the process for a general quadratic equation: Start with the quadratic equation in standard form: ax^2 + bx + c = 0 If a≠1, divide both sides by a to make the coefficient of x^2 equal to 1: x^2 + (b/a)x + (c/a) = 0 Move the constant term to the other side: x^2 + (b/a)x = -(c/a)Find the value to complete the square: Take half of the coefficient of x, square it, and add it to both sides. The value to add is 〖1/2〗^2. x^2 + (b/a)x + 〖(b/2a)〗^2 = -(c/a) + 〖(b/2a)〗^2 Rewrite the left side as a squared binomial: (x + (b/2a)^2)*(x + (b/2a)^2) or [x + (b/2a)^2]^2 Solve for x by taking the square root of both sides and isolating x.In this case: Completing the Square for x^2 - 10x = 17 Rearrange to standard form: x^2 - 10x - 17 = 0 We see that a=1. So dividing by a (1) gives us: x^2 - 10x - 17 = 0Move the constant to the other side: x^2 - 10x = 17Take half of the coefficient of x [-5], square it [25], and add it to both sides. The value to add is 25. x^2 - 10x + 25 = 17 + 25Now the left side is a perfect square, so rewrite the expression: (x - 5)^2 = 42Take the square root of both sides; x - 5 = ± (42)^(1/2) The two possible answers are: x = 5 + (42)^(1/2) x = 5 - (42)^(1/2)

Answered by rspill6 | 2025-07-23