The equation `y=ax^2+bx+c` is a means of describing the quadratic function.

If a quadratic function is equal to zero, the result will be a quadratic equation with roots, `x` . The x-values are the roots (or zeros) of `f(x) = 0.`

The graph of a quadratic function is called a parabola. In the equation, a change in `a` will change the shape of the parabola.

A change in `b` will change the placement of the vertex or turning point and

a change in `c` will move the parabola up or down.

Distance, speed and time, etc can be measured using quadratic equations. A practical real life example is throwing a ball. A parabola will show you how high the ball goes and when it will hit the ground. Quadratic functions can also be used to determine profit and the best price for maximum profit. a b and c therefore measure changes.

The black graph represents `y=x^2` where `b=0` and `c=0`

The red graph represents `y=x^2 +2` where `b=0`

the green graph represents `y=x^2 +2x +2`

**Ans: The equation `y=ax^2+bx+c` is important as it is necessary for practical applications in many instances of mathematics, science and technology. **

` ` `y=ax^2 + bx +c`is the original function for a parabola. You can change the shape and location of this by increasing the a, b, and c values.

This equation can also be factored to the form:`y = (x+-n)(x+-m)` .

The family function of a parabola is `y=x^2` where the vertex is on the origin.

You can also use the quadratic equation from this equation:

`x = (-b +- sqrt(b^2 -4ac))/(2a)`

You use this equation when you can't factor easily.

the equation describes a quadratic function and from it the quadratic formula can be derived.

`y = ax^2 + bx +c` is important because it is known as the quadratic formula. The quadratic formula is used when trying to find the variable x. The a,b, and c variable can be used to find x through the quadratic formula `x = (-b+-(b^2 -4ac))/(2a)` . Where you would just plug in a,b, and c in order to find x

It is known as the quadratic formula, this formula is used to find missing variable and can help with a lot of struggling when you cant find what a variable might be.

The equation you just described is called the quadratic equation. Its so very important because it enables people to solve functions to the second degree without factoring (many equations can't be factored so this method is especially useful for them) and without tediously completing the square (which has its own benefits especially in finding integrals in calculus). Now, if the question is: why is it important we're able to solve such equations; just look at at any STEM career/application. (Science, Technology, Engineering, Mathematics).