Question

What is the quadratic formula?

Answer 1

Hint: A polynomial of the form \[a{x^2} + bx + c,a \ne 0\] and when we equate this polynomial to zero i.e.\[a{x^2} + bx + c = 0,a \ne 0\], then the equation/formula we get is known as quadratic equation or quadratic formula.

Complete step-by-step solution:
Let us take an example of blackboard of our school which usually has rectangular shape,

Now we have two information regarding the blackboard, first is the area of blackboard is \[10{m^2}\] and the length is one meter more than twice its breadth
As we know that,
Area of rectangle \[ = length \times breadth\]
So we have,
Area of rectangle \[ = (2x + 1) \times (x)\]
\[(2x + 1) \times (x) = 10\] (Given)
After multiplication we get,
\[2{x^2} + x = 10\]
Therefore, \[2{x^2} + x - 10 = 0\]
So, the breadth of the blackboard should satisfy the equation \[2{x^2} + x - 10 = 0\] which is a quadratic equation.
A quadratic equation is an equation of the form \[a{x^2} + bx + c = 0\], where a, b, c are real numbers and \[a \ne 0\]. For example, \[2{x^2} + x - 10 = 0\] is a quadratic equation. We can also say that, any equation of the form \[p(x) = 0\], where \[p(x)\] is a polynomial of degree $2$ , is a quadratic equation. But by writing the terms of \[p(x)\] in descending order of their degrees, we get the standard form of the equation. \[a{x^2} + bx + c = 0\] , \[a \ne 0\] is called the standard form of a quadratic equation.
The Solution of quadratic equation and relation between roots and coefficient:
> The solutions of the quadratic equation, \[a{x^2} + bx + c = 0\] is given by \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
> The expression \[{b^2} - 4ac = D\] is called the discriminant of the quadratic equation.
> If \[\alpha {\text{ }}\& {\text{ }}\beta \] are the roots of the quadratic equation \[a{x^2} + bx + c = 0\] then;
1) \[\alpha + \beta = - \dfrac{b}{a}\]
2) \[\alpha \beta = \dfrac{c}{a}\]
3) \[\left| {\alpha - \beta } \right| = \dfrac{{\sqrt D }}{{\left| a \right|}}\]
> Quadratic equation whose roots are \[\alpha {\text{ }}\& {\text{ }}\beta \] is \[(x - \alpha )(x - \beta ) = 0\] i.e.
 \[{x^2} - (\alpha + \beta )x + \alpha \beta = 0\] i.e. \[{x^2} - \] (sum of roots) \[x\]\[ + \] product of roots \[ = 0\]
Nature of Roots: Consider the quadratic equation \[a{x^2} + bx + c = 0\] where \[a,b,c \in R\& a \ne 0\] then;
> \[D > 0 \Leftrightarrow \] Roots are real & distinct (unequal).
> \[D = 0 \Leftrightarrow \] Roots are real & coincident (equal).
> \[D < 0 \Leftrightarrow \] Roots are imaginary.

Note: To solve a quadratic equation by factoring,
> Rearrange the equation and take all terms on one side of the equal sign and zero on the other side.
> Factorize the equation.
> Take each factor equal to zero and solve each equation.
> Check the calculated answer by putting it in the original equation.