Standard form of a quadratic equation and coefficients.

The standard form of a quadratic equation:

The standard form of a quadratic equation is given by $f\left( x\right) =ax^{2}+bx+c .$ It contains three terms with a decreasing power of “x”.

Here, “a” is the coefficient of $x^ {2}$ which is generally called as leading coefficient,“b” is the coefficient of “x” and the “c” is called as the constant term.

The leading coefficient “a” can not be “0” in quadratic standard form of equation.

Identification of the coefficients of quadratic is a key part in further topics of the quadratic expression. It is primarily used in the quadratic formula, finding the vertex of a quadratic equation and factorization for finding “x” intercepts of a quadratic equation.

1)Worked out examples:-

Find the coefficients a,b,c of the given the standard form of a quadratic equation.

$i) 2x^{2}+4x +5$

2) Strategy Step-by-Step:-

Step1) First of all, we take the standard form of the quadratic equation $i) ax^{2}+bx +c$

Step2) Compare the standard form of the quadratic with the given equation One-to-one as shown below.

Step 3) Identify the corresponding coefficients a=2 ,b=4,c=5.

Step 4) If the quadratic standard form has missing $x^{2}$ ,x or constant c terms, we make the corresponding missing coefficient as a “0”.

3) Practice Problems

Identify the coefficients a,b,c of the quadratics given.

$i) 0.5x^{2}+3x +5$

$ii) -1x^{2}+4.$

$iii) y^{2}+2y.$

$iv)4x+ 5.$

$v)z^{2}+2z+5.$

4) Solutions:-

i) $0.5x^{2}+3x +5$ .On comparison with the standard form the quadratic equation $f\left( x\right) =ax^{2}+bx+c .$ , we observe that there is a 0.5 before the “x^2” term and 3 before the “x” term and a constant c.

Therefore on comparision we conclude that a= 0.5 ,b=3 and c=5.

ii) $-x^{2}+4$ .On comparison with the standard form the quadratic equation $f\left( x\right) =ax^{2}+bx+c .$ , we observe that there is a “-1” before the “x^2” term and there is No “x” term given.

Since there is no “x” term available, the corresponding coefficient “b” of variable “x” is to be considered as a “0”.

Therefore one comparison ,as shown in the picture below, with the standard form, we can say that a= -1,b=0 and c=4.

iii) $y^{2}+2y.$ . The very first observation is that this is a quadratic with “Y” as a variable. But still, this equation can be compared with the standard form of the quadratic $f\left( x\right) =ax^{2}+bx+c .$ .

There is no restriction on the variable of a quadratic equation. We can choose variable of any alphabet.
Therefore on a comparison, we observe that we have y^2 alone and there is no number before y^2. Therefore we need to take the default coefficient as “1”.

The coefficient of the variable “y” is “2”.Therefore, on comparison, the value b=2. Lastly there is NO constant number in this quadratic. So the constant variable c=0.

iv) $4x+ 5.$ .Given expression is in variable “x” and it does not have the x^2 term. Hence, given expression is not a quadratic expression anymore.

Standard form of a Quadratic and identifying the coefficients in a quadratic.

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