how to solve quadratic equation | Zero math

how to solve quadratic equation | trigonometry problem solving | Zero math
There are three main methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula. Let's discuss each method in detail.
Method 1: Factoring
Factoring is a common method for solving quadratic equations when the quadratic expression can be factored into two linear expressions. To factor a quadratic expression, you should first identify the greatest common factor (GCD) of all the terms. If the GCD is not 1, you can factor it out and factor the remaining expression.
Once you have factored the expression into two linear expressions, you can set each linear expression equal to zero and solve for the value of x. For example, consider the quadratic equation:
x^2 + 5x + 6 = 0
This equation can be factored as:
(x + 2)(x + 3) = 0
Setting each linear expression equal to zero, we get:
x + 2 = 0 or x + 3 = 0
Solving for x, we get:
x = -2 or x = -3
Therefore, the solutions to the quadratic equation x^2 + 5x + 6 = 0 are x = -2 and x = -3.
Method 2: Completing the Square
Completing the square is a method for solving quadratic equations when the quadratic expression cannot be easily factored. The goal of completing the square is to rewrite the quadratic expression in the form of a perfect square trinomial.
To complete the square, you should first move the constant term to the right side of the equation. Then, divide both sides of the equation by the coefficient of the x^2 term. Next, take half of the coefficient of the x term, square it, and add it to both sides of the equation.
Finally, rewrite the left side of the equation as a perfect square trinomial. By taking the square root of both sides of the equation, you can solve for x.
For example, consider the quadratic equation:
x^2 + 4x - 5 = 0
Completing the square, we get:
(x^2 + 4x) = 5
(x^2 + 4x + 4) = 5 + 4
(x + 2)^2 = 9
x + 2 = ±3
x = -2 ± 3
x = 1 or x = -5
Therefore, the solutions to the quadratic equation x^2 + 4x - 5 = 0 are x = 1 and x = -5.
Method 3: Quadratic Formula
The quadratic formula is a general formula that can be used to solve any quadratic equation. The formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
To use the quadratic formula, you should first identify the values of a, b, and c. Then, substitute the values of a, b, and c into the formula and simplify.
For example, consider the quadratic equation:
2x^2 + 3x - 2 = 0
In this equation, a = 2, b = 3, and c = -2. Substituting these values into the quadratic formula, we get:
x = (-3 ± √(3^2 - 4 * 2 * -2)) / 2 * 2
x = (-3 ± √25) / 4
x = (-3 ± 5) / 4
x = 1/2 or x = -2
Therefore, the solutions to the quadratic equation 2x^2 + 3x - 2 = 0 are x = 1/2 and x = -2.
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