Definition of Quadratic functions

Quadratic functions are used in algebra that are polynomial functions, that contaions one or more variables. In a quadratic function, the higest degree term is of second degree. It is also known as quadratic polynomial or polynomial of degree 2 and also simply quadratic. An example of quadratic function can be: f (x,y,z) = ax² + by² +cz² + dxy + exz + fyz + gx + hy + iz + j

This quadratic function contains, variables like x, y and z and also contaions some exclusive terms like  x², y², z², xy, xz, yz, x, y, z, and a constant which is j. The second degree terms for the coefficients a, b, c, d, e, or f are non zero.

An example of a single variable or univariate quadratic function will be: f (x,y) = ax2 + bx + c , a "≠" 0

Here, c is a constant.

The graph that is being generated using a quadratic function is known as parabola. The parabolas have a basic “U” shape. Every parabola is symmetric with respect to the axis of symmetry. The points at which the parabola intersects the axis of symmetry is known as the vertex of the parabola.

The steps that are required to solve a quadratic equation with the help of quadratic formula with examples equaltions

Quadratic formula:

Example 1: Solve then quadratic equation x² – 8x + 14 = 0

The first step is to identify a, b and c from the given equation. In this  case  a = 1, b = –8, and c = 14.  

The next step is to substitute the values in the quadratic formula,

Now, solve the equation,

Simplify the quation, if only there are chances of simplification,

Solving a quadratic equation by factoring

The steps to do this correctly are:

1. At first, the equation should be arranged correctly. All the terms should be moved to the left hand side which will be equal to the value zero.
2. After the rearrangement, the factoring strategy should be used to factor the problem.
3. The next step is to use the zero product property, which sets each factor that contain a variable equal to the value zero.
4. The final step is to solve the factors that were set to zero and find the value of x.

Example 1: Solve the quadratic equation x² + 16 = 10x

Step 1: x² – 10x + 16 = 0

Step 2: Use factoring strategy

(x – 2)(x - 8) = 0

Step 3: Use the zero product property

 x – 2 = 0 or x – 8 = 0

Step 4: Solve for each factor

 x = 2 or x = 8

Example 2: Solve the quadratic equation 50x² = 72

Step 1: Rearrange the equation 50x² – 72 = 0

Step 2: Use the factoring strategy

2(25x² - 36) = 0

2(5x + 6)(5x - 6) = 0

Step 3: Use zero product property

5x + 6 = 0 or 5x – 6 = 0

Step 4: Solve for each factor

5x = -6 or x = -1

x = - 6/5 or x = 6/5