Quadractic Functions?
How do you solve them?
1. Factoring. Only special ones factor
2. Completing the square, works every time
3. the Quadratic Formula, works every time.
I need to know what the answer is to: x^2-__+49/4=( )^2
Quadratic functions are a type of polynomial function with a degree of 2. They have the general form: f(x) = ax^2 + bx + c, where a, b, and c are constants. To solve a quadratic function, you can use several techniques, such as factoring, completing the square, or using the quadratic formula.
1. Factoring: If the quadratic expression can be factored, you can set it equal to zero and solve for x by applying the zero-product property. For example, let's consider the quadratic function x^2 + 5x + 6 = 0. You would factor it as (x + 2)(x + 3) = 0 and then set each factor to zero: x + 2 = 0 and x + 3 = 0. Solving these, you find x = -2 and x = -3 as the solutions.
2. Completing the Square: This method involves manipulating the quadratic expression to create a perfect square trinomial. Rewrite the quadratic function in the form: a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. Solve by setting this expression equal to zero and isolating x. Let's use the quadratic function x^2 + 4x - 5 = 0 as an example. Completing the square, you get (x + 2)^2 - 9 = 0. Setting this equal to zero, you have (x + 2)^2 = 9. Solve for x to find x = -5 and x = 1 as the solutions.
3. Quadratic Formula: The quadratic formula is a convenient way to find the solutions to any quadratic function. The formula is x = [-b ± √(b^2 - 4ac)] / 2a, where a, b, and c are the coefficients of the quadratic function. Using the same example, x^2 + 4x - 5 = 0, you can apply the quadratic formula to find x = (-4 ± √(4^2 - 4(1)(-5))) / (2(1)). Simplifying, you get x = (-4 ± √(16 + 20)) / 2, which gives x = (-4 ± √36) / 2. This leads to x = (-4 ± 6) / 2, resulting in x = -5 and x = 1 as the solutions.
Remember to always check your solutions by substituting them back into the original quadratic equation to ensure they satisfy the equation.