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# How do you solve the equation ${x^2} - 3x - 2 = 0$ using the quadratic formula?

Last updated date: 19th Jun 2024
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Hint: This question is from the topic of solutions of quadratic equations. We have to solve this question using the quadratic formula. To solve this we need to know the quadratic formula and discriminant of a quadratic equation. Discriminant of a quadratic equation gives details about the nature of the roots of a quadratic equation. This question is very easy. You just need to apply the formula. Try once by yourself before looking at a complete solution.

Complete step by step solution:
Let us try to solve this question in which we have to find the roots of a quadratic equation ${x^2} - 3x - 2 = 0$ using the quadratic formula. Quadratic formula is given by $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ for any general quadratic equation $a{x^2} + bx + c = 0$ where ${b^2} - 4ac$ is called the discriminant of quadratic equation, it tells the nature of roots of quadratic equation. Here are conditions:
1) Two distinct real roots, if ${b^2} - 4ac > 0$
2) Two equal real roots, if ${b^2} - 4ac = 0$
3) No real roots, if ${b^2} - 4ac < 0$
In the given quadratic equation we have,
$a = 1 \\ b = - 3 \\ c = - 2 \\$
Discriminant of the quadratic equation is
${b^2} - 4ac = {(3)^2} - 4 \cdot 1 \cdot ( - 2) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 9 + 8 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 17 > 0 \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\$
Hence the given quadratic equation has two distinct real roots, because discriminant is greater than $0$.
Now putting values of $a,b$ and $c$ in quadratic formula we get,
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{ - ( - 3) \pm \sqrt {{{( - 3)}^2} - 4 \cdot 1 \cdot ( - 2)} }}{2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \dfrac{{3 \pm \sqrt {9 + 8} }}{2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \dfrac{{3 \pm \sqrt {17} }}{2} \\$
Hence the root of quadratic equation ${x^2} - 3x - 2 = 0$ are $x = \dfrac{{3 + \sqrt {17} }}{2}$and$x = \dfrac{{3 - \sqrt {17} }}{2}$.

Note: To solve questions in which you are asked to find the roots of quadratic equations by quadratic formula you must need to know the formula. We can also solve this by using other methods of finding roots of a quadratic equation such as completing square method and factor method.