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Find the sum and product of zeros of the polynomial\[7{x^2} + 5x - 2 = 0\].

Last updated date: 20th Jun 2024
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Hint: A polynomial is an expression that contains the variables and coefficients which involves operations like addition, subtraction, multiplication. Quadratic equations are the equation that contains at least one squared variable, which is equal to zero. Quadratic equations are useful in our daily life it is used to calculate areas, speed of the objects, projection, etc.

The quadratic equation is given as \[a{x^2} + bx + c = 0\]this is the basic equation which contains a squared variable \[x\]and three constants a, b and c. The value of the \[x\] in the equation which makes the equation true is known as the roots of the equation. The numbers of roots in the quadratic equations are two as the highest power on the variable of the equation is x. The roots of the equation are given by the formula\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], where \[{b^2} - 4ac\]tells the nature of the solution.
In the quadratic equation, \[a{x^2} + bx + c = 0\]the sum of the roots is given by \[ - \dfrac{b}{a}\]whereas their product is given \[\dfrac{c}{a}\].

Complete step by step answer:
In the given quadratic equation \[7{x^2} + 5x - 2 = 0\]--- (i)
Let us assume that the zeros or the roots of the equation to be \[\alpha \] and\[\beta \].
As we know, the sum of the roots of the equation is given as\[\alpha + \beta = - \dfrac{b}{a}\]
So by comparing equation - (i) with the general quadratic equation, \[a{x^2} + bx + c = 0\] we can write
 \[\alpha + \beta = - \dfrac{b}{a} = - \dfrac{5}{7}\]
And the product of the roots of a quadratic equation is given as \[\alpha \beta = \dfrac{c}{a}\]
So compare equation (i) with the general quadratic equation \[a{x^2} + bx + c = 0\]we get
\[\alpha \beta = \dfrac{c}{a} = \dfrac{{ - 2}}{7} = - \dfrac{2}{7}\]
Hence the sum and the product of a quadratic equation are \[\alpha + \beta = - \dfrac{5}{7}\]and \[\alpha \beta = - \dfrac{2}{7}\] respectively.

Note: In the quadratic equation if \[{b^2} - 4ac > 0\]the equation will have two real roots. If it is equal, \[{b^2} - 4ac = 0\]then the equation will have only one real root and when \[{b^2} - 4ac < 0\]then the root is in complex form.