Mastering the Methods: Factoring, Quadratic Formula, and Completing the Square for Solving Quadratic Equations

quadratic

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a is not equal to zero

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a is not equal to zero. The highest power in a quadratic equation is 2.

To solve quadratic equations, we can use factoring, the quadratic formula, or completing the square.

1. Factoring:
First, we attempt to factorize the quadratic equation if possible. For example, let’s consider the equation x^2 + 5x + 6 = 0.
We need to find two numbers whose product is equal to ac (in this case, 6) and whose sum is b (in this case, 5). The numbers are 2 and 3.
So, we can rewrite the equation as (x + 2)(x + 3) = 0.
Now, we set each factor equal to zero and solve for x.
x + 2 = 0 -> x = -2
x + 3 = 0 -> x = -3
Therefore, the solutions (or roots) of the quadratic equation x^2 + 5x + 6 = 0 are x = -2 and x = -3.

2. Quadratic Formula:
If factoring is not possible or convenient, we can use the quadratic formula to solve quadratic equations.
The quadratic formula is given by x = (-b ± √(b^2 – 4ac)) / (2a).
For example, let’s solve the equation 2x^2 – 5x + 3 = 0 using the quadratic formula.
Here, a = 2, b = -5, and c = 3.
Plugging these values into the quadratic formula, we get:
x = (-(-5) ± √((-5)^2 – 4*2*3)) / (2*2)
Simplifying further:
x = (5 ± √(25 – 24)) / 4
x = (5 ± 1) / 4
This gives us two solutions:
x = (5 + 1) / 4 = 6 / 4 = 3/2
x = (5 – 1) / 4 = 4 / 4 = 1
Therefore, the solutions of the quadratic equation 2x^2 – 5x + 3 = 0 are x = 3/2 and x = 1.

3. Completing the Square:
Completing the square is another method to solve quadratic equations. It involves manipulating the equation to express it in the form (x – h)^2 = k, where h and k are constants.
For example, let’s solve the equation x^2 – 6x + 8 = 0 using completing the square.
First, rearrange the equation by moving the constant term to the right side:
x^2 – 6x = -8
Next, take half of the coefficient of x (which is -6) and square it. Add this value to both sides of the equation:
x^2 – 6x + 9 = -8 + 9
(x – 3)^2 = 1
Now, taking the square root of both sides, we get:
x – 3 = ± √1
x – 3 = ± 1
Solving for x:
x = 3 + 1 = 4
x = 3 – 1 = 2
Therefore, the solutions of the quadratic equation x^2 – 6x + 8 = 0 are x = 4 and x = 2.

These are the main methods used to solve quadratic equations: factoring, quadratic formula, and completing the square. Depending on the specific equation, one method may be more convenient or appropriate than the others.

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