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In Algebra help, An equation is an equality which is satisfied only by some particular values of the variables occurring in it. Solving equations is a very important topic covered under algebra.

**QUADRATIC EQUATIONs**:

An equation of the form ax² + bx + c = 0 where a, b, c are certain numbers, and a $\neq$ 0 is called a quadratic equations. The number a, b, c are called the coefficients of the quadratic equation and the number b² - 4ac is called its discriminant. Discriminant of a quadratic equation is usually denoted by D.

A quadratic equation in many cases can be factored in the form

(x – x_{1})(x – x_{2}) = 0, where x_{1} and x_{2} are the roots, and the roots can be found by equating each factor to 0. But also in many cases factoring found to be not possible. In such situations, a formula has been derived to find the roots in general. It states that, if ax^{2} + bx + c = 0 and if x_{1} and x_{2} are the roots, then,

$x_1$ = $\frac{-b + \sqrt{b^2-4ac}}{2a}$

$x_2$ = $\frac{-b - \sqrt{b^2-4ac}}{2a}$

The term b^{2 }– 4ac is called as the discriminant, D, of the quadratic equation and it decides the nature of the root. Rewriting the roots in discriminant form,

$x_2$ = $\frac{-b - \sqrt{D}}{2a}$

Now, we find the followings.

If D < 0, the radicand is negative and $\sqrt{D}$ is imaginary and has two values. Therefore, if D < 0, the roots are imaginary and different.

If D = 0, the radicand is 0 and $\sqrt{D}$ becomes 0. Therefore, if D = 0, the roots are real and identical.

If D > 0, the radicand is positive and $\sqrt{D}$ is real and has two values. Therefore, if D > 0, the roots are real and different.

We have seen that the roots of a quadratic equation are,

$x_1$ = $\frac{-b + \sqrt{b^2-4ac}}{2a}$$x_2$ = $\frac{-b - \sqrt{b^2-4ac}}{2a}$

By addition both the roots,

$x_1 + x_2$ _{ }= $\frac{-b + \sqrt{b^2-4ac}}{2a}$ + $\frac{-b - \sqrt{b^2-4ac}}{2a}$ = $\frac{-b}{a}$

Therefore, the sum of the two roots is $\frac{-b}{a}$ and this is also the x-coordinate of the vertex of the function. Also, therefore, x = (-b/a) represents the equation of symmetry.

Now by multiplication,

$x_1 . x_2$ = $\frac{-b + \sqrt{b^2-4ac}}{2a}$ . $\frac{-b - \sqrt{b^2-4ac}}{2a}$ = $\frac{c}{a}$

Therefore, the product of the two roots is $\frac{c}{a}$.