How do I complete the square for a quadratic equation? I'm not sure how to do it.
eg
x^2 -8x + 7
If the coefficient of the square term is 1 and the middle term is even, they are easy
take 1/2 of the coefficient of the x term, square it, then add and subtract it
x^2 - 8x + 16 - 16 + 7
your first 3 terms are now a perfect square, express it that way
(x-4)^2 - 16 + 7 simplify the constants
for a final answer of
(x-4)^2 - 9
now check your answer by expanding this
the worst kind are this kind
4x^2 + 5x + 1
factor out the 4 from the first 2 terms
4(x^2 + (5/4)x ) + 1 notice I left the +1 alone
now repeat the steps from my first example
4(x^2 + (5/4)x + 25/64 - 25/64) + 1
= 4(x + 5/8)^2 - 25/64) + 1 multiply out the number in front
= 4(x+5/8)^2 - 25/16 + 1 adding up the constants
= 4(x+5/8)^2 - 9/16
Write in in the form
x^2 + bx = -c
where -c is a constant.
Then add b^2/4 to both sides. The left side becomes [x +(b/2)]^2 and the right side becomes (b^2/4) - c
Then take the square root of both sides.
To complete the square for a quadratic equation, you can follow these steps:
1. Start with a quadratic equation in the general form: ax^2 + bx + c = 0, where a, b, and c are constants.
2. Divide both sides of the equation by a (the coefficient of x^2) to make the coefficient of x^2 equal to 1. This step ensures that the quadratic equation is in the standard form.
3. Rearrange the equation so that the x terms are grouped together on one side, and the constant term is on the other side.
4. Subtract the constant term (c/a) from both sides of the equation.
5. Take half of the coefficient of x (b/2a), square it [(b/2a)^2], and add it to both sides of the equation.
By following these steps, you have "completed the square" for the quadratic equation. This process helps in solving quadratic equations, finding the vertex form, and graphing the parabola associated with the quadratic equation.
Completing the square is often used to convert a quadratic equation into a perfect square trinomial, which allows you to easily factor the equation or solve for the roots.
If you provide a specific quadratic equation, I can walk you through the process with an example.