Find the roots of the following quadratic equation, if they exist, using the quadratic formula of Shridhar Acharya. \[9{x^2} + 7x - 2 = 0\]
ANSWER
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Hint: According to the Shridhar Acharya formula to find the roots of the quadratic equation \[a{x^2} + bx + c = 0\], roots of this quadratic equation is calculated as \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].
Complete step-by-step answer: As we know that we had to find the roots of the given quadratic equation using the formula given by Shridhar Acharya. So, as we know that according to the Shridhar Acharya the roots of the quadratic equation \[a{x^2} + bx + c = 0\] exist only if the value of \[{b^2} - 4ac\] is greater than or equal to zero. So, let us find the value of \[{b^2} - 4ac\] for the given quadratic equation. So, first we have to find the value of a, b and c for the quadratic equation \[9{x^2} + 7x - 2 = 0\] As we know that b is the coefficient of x in any quadratic equation. So, b = 7 And a is the coefficient of \[{x^2}\]. So, a = 9. And c is the constant term of any quadratic equation. So, c = – 2. Now, \[{b^2} - 4ac\] = \[{\left( 7 \right)^2} - 4\left( 9 \right)\left( { - 2} \right)\] = 49 + 72 = 121. So, as we can see that the value of \[{b^2} - 4ac\] is 121 which is greater than 0. So, the roots of the given quadratic equation exist. Now as we know that the roots of the quadratic equation are calculated as \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. So, roots of the given quadratic equation are \[\dfrac{{ - 7 \pm \sqrt {121} }}{{2\left( 9 \right)}} = \dfrac{{ - 7 \pm 11}}{{18}}\] = \[\dfrac{{ - 7 + 11}}{{18}}\] and \[\dfrac{{ - 7 - 11}}{{18}}\] = \[\dfrac{2}{9}\] and \[ - 1\]. Hence, the roots of the quadratic equation exist and are equal to \[\dfrac{2}{9}\] and \[ - 1\].
Note: Whenever we come up with this type of problem we have to first check whether the roots of the given quadratic equation exist or not by using the formula \[{b^2} - 4ac\]. If the value of \[{b^2} - 4ac\] is less than zero then the roots does not exist or we can say that the roots are unreal, if the value of \[{b^2} - 4ac\] is equal to zero then the roots are equal and if the value of \[{b^2} - 4ac\] is greater than zero then the roots are real and different. After that we can apply the formula \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] to find the roots of the given quadratic equation, if they exist.