How to Solve Quadratic Equations: Factoring, Quadratic Formula, and Completing the Square

quadratic

A quadratic equation is a polynomial equation of degree 2

A quadratic equation is a polynomial equation of degree 2. It can be written in the form:

ax^2 + bx + c = 0

Where a, b, and c are constants, and x represents the variable.

To solve a quadratic equation, there are a few different methods you can use:

1. Factoring: If the equation is easily factorable, you can factor it into two binomials and set each factor equal to zero. Solve for x in each case. For example, if you have the equation x^2 + 5x + 6 = 0, you can factor it as (x + 2)(x + 3) = 0. Then, you can set each factor equal to zero and solve for x: x + 2 = 0 and x + 3 = 0, which gives you x = -2 and x = -3 as the solutions.

2. Using the Quadratic Formula: If factoring is not possible, you can use the quadratic formula to solve the equation. The quadratic formula is given by:

x = (-b ± sqrt(b^2 – 4ac)) / (2a)

Plugging in the values of a, b, and c from your quadratic equation into the quadratic formula, you can solve for x. For example, if you have the equation 2x^2 – 3x – 2 = 0, you can use the quadratic formula to find the solutions:

x = (-(-3) ± sqrt((-3)^2 – 4(2)(-2))) / (2(2))
= (3 ± sqrt(9 + 16)) / 4
= (3 ± sqrt(25)) / 4
= (3 ± 5) / 4

This gives you two solutions: x = (3 + 5) / 4 = 2 and x = (3 – 5) / 4 = -1/2.

3. Completing the Square: Another method for solving quadratic equations is by completing the square. This method involves manipulating the equation to get it in the form of a perfect square trinomial. Once you have a perfect square trinomial, you can solve for x by taking the square root of both sides of the equation. However, the quadratic formula is generally easier to use and is more commonly taught.

Note that a quadratic equation can have zero, one, or two real solutions, depending on the discriminant (the value inside the square root of the quadratic formula). If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has one real solution (also called a double root). If it is negative, the equation has no real solutions, but rather complex solutions.

I hope this explanation helps! Let me know if you have any further questions.

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