Completing the Square: Solving -x^2 + 4x and Finding the Vertex

  • Thread starter Miike012
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In summary, to complete the square for the expression -x^2 + 4x, you need to add +4 inside the parentheses to balance the minus sign out front. This will result in the vertex being at (2, -4), which may seem incorrect but is actually correct due to the negative coefficient of x^2.
  • #1
Miike012
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Problem: -x^2 + 4x

- ( x^2 - 4x + (4/2)^2 ) - 4
= - (x - 2) - 4

Vertex is at (2, -4)

but the vertex is obviously at (2,4)...
what did I do wrong.
 
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  • #2
Miike012 said:
Problem: -x^2 + 4x

- ( x^2 - 4x + (4/2)^2 ) - 4
The error is here. You have actually added -4 inside the parentheses to complete the square, so to keep the expression equal, you need to add + 4.
Miike012 said:
= - (x - 2) - 4

Vertex is at (2, -4)

but the vertex is obviously at (2,4)...
what did I do wrong.
 
  • #3
how did I add -4... then the expression would be x^2 - 4x - 4 which is not a perfect square...? Does it have something to do with x^2 being negative?
 
  • #4
Let's go through the steps.

-x2 + 4x
= -(x2 - 4x)
= -(x2 - 4x + 4) + 4
I put in +4 inside the parentheses, but due to the minus sign out front, I have actually added -4, so to balance, I have to add + 4.

= -(x - 2)2 + 4
 
  • #5
Another way of looking at it is to do both addition and subtraction inside the parentheses:
[tex]-(x^2- 4x)= -(x^2- 4x+ 4- 4)= -(x^2- 4x+ 4)-(-4)= -(x- 2)^2+ 4[/tex]
 
  • #6
Mark44 said:
Let's go through the steps.

-x2 + 4x
= -(x2 - 4x)
= -(x2 - 4x + 4) + 4
I put in +4 inside the parentheses, but due to the minus sign out front, I have actually added -4, so to balance, I have to add + 4.

= -(x - 2)2 + 4

or if you like

-x2 + 4x
= -[x2 - 4x]
= -[x2 - 4x + 4 - 4]
= -[(x - 2)2 - 4]
= -(x - 2)2 + 4
 

What is the purpose of completing the square when solving a quadratic equation?

Completing the square is a method used to solve quadratic equations, which are equations in the form of ax^2 + bx + c = 0. It involves manipulating the equation to create a perfect square trinomial, which makes it easier to find the solutions or roots of the equation.

How do you complete the square to solve a quadratic equation?

To complete the square, take the coefficient of x, divide it by 2, and square the result. Add this value to both sides of the equation, and factor the perfect square trinomial on the left side. This will result in an equation in the form of (x + a)^2 = b, where a and b are constants. You can then take the square root of both sides to solve for x.

What is the role of the constant term when completing the square?

The constant term, or the term without an x, is a crucial component when completing the square. It is used to add and subtract values on both sides of the equation to create a perfect square trinomial. It also helps determine the y-coordinate of the vertex, which is the highest or lowest point on the graph of the quadratic function.

What is the vertex and why is it important in solving quadratic equations?

The vertex is the point on the graph of a quadratic function where the graph changes direction, either from increasing to decreasing or vice versa. It is represented by the coordinates (h, k), where h is the x-coordinate and k is the y-coordinate. The vertex is important because it gives us information about the maximum or minimum value of the function, which can be useful in real-life applications.

Can completing the square be used to solve any quadratic equation?

Yes, completing the square can be used to solve any quadratic equation. However, it is not always the most efficient method, especially when the coefficient of x^2 is not equal to 1. In those cases, using the quadratic formula may be a better option. Completing the square is also useful for graphing quadratic functions and finding the maximum or minimum values of the function.

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