Let us assume that a man ‘a’ got “p * p” number of houses, a woman ‘b’ has “q” number of houses, and another girl ‘c’ has only one house. If we wish to sum up all three person’s houses along with their respective amounts, is there any way to represent this scenario in mathematical terms? Well, let us do this:
Person 1: a p * p = ap2
Person 2: b q
Person 3: c * 1
Here, a, p, b, q, c are variables.
Summing up these terms formed is:
ap2 + bq + c
In another form, we have: y = ap2 + bq + c.
Here, we wrote p * p as p2, which shows that this equation we formed is Quadratic. Here, “Quad” means two and a quadratic equation means an equation having a power of 2. Please note that a, b, and c are constants, and p, q are variables. For instance 4p2 + 5q + 3.
Now, if it happens that on substituting values of p and q, in which either of these is negative, then the equation becomes equal to zero. So we write the equation as;
ap2 + bq + c = 0….(1)
4p2 + 5q + 3 = 0
Here, eq (1) is the standard form of a Quadratic Equation.
How Do We Find the Solution of a Given Quadratic Equation?
Let’s learn how do you find the solution of a quadratic equation with an equation. It would make it easier for you to understand it.
Suppose there is an equation x2+4x + 4 = 0, how do we find the roots of this equation? Let us understand this.
x2+4x + 4 can be written as: x2 + 2 (x + 2)
Now, finding out the common terms:
= (x + 2) ( x + 2) = 0
Now, we have (x + 2) = 0 and (x + 2) =0
x + 2 =0
Bringing ‘2’ on RHS changes its sign, we get x = – 2. As there are two (x + 2), we have two times x = – 2. So, this is how we find roots in a Quadratic Equation.
Easiest Way to Factorise Any Quadratic Equation
Assume that we have an equation 3x2 + 4x -15 =0and now we have to factorize this equation, so how do we do this? Well, let us understand this below.
There is one of the famous Quadratic Formula, p = 3, q = 4, and c = – 15, so putting these values we have:
Writing factors of each term below the equation as;
3x2 + 4x -15 =0
1 x 3 + sign
3 x 5 – sign
Now, cross-multiplying as:
We get, 1x * (- 5) = – 5x, snd 3x * (3) = + 9x. So, we get our desired equation as (x + 3) (3x – 5).
What is Quadratic Function?
Well, we know that a quadratic equation is in the form of ap2 + bp + c = 0. However, when we want to represent any quadratic function, we have the following form:
f (p) = ap2 + bp + c
Here, a, b, and c are constants not equal to zero. Also, the graph of this equation is a parabola. A parabola has a vertex. So, how does a vertex form? Well, when a parabola intersects an axis of symmetry, and the point where parabola intersects is a vertex.
Since we know that a point is formed of two points, which means if there are two points in a plane, then there is only one that connects two points, the same happens in the case of quadratic function.
For instance, there is an equation for parabola: m = n (x – h)2 + k, where h and k are vertices of this parabola.
Now, let us consider an equation for parabola: f (p) = 3p2 + 12p – 12, let us find the p-coordinate of the vertex:
Considering y = f (p) = 3p2 + 12p – 12
We know that formula for finding the p-coordinate for vertex is:
Here, in the above equation a = 3 and b = 12
– 12/2 * 3 = – 2
So, finding f (p) by substituting the value of ‘p’ in the given equation:
3* (2)2 + 12 * ( – 2) – 12
So, we get f (p) = – 24
Hence, the vertex of parabola 3p2 + 12p – 12 is (- 2, -24).
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