i.e., if a + bi is a root then a - bi is also a root. Note: A quadratic equation can never have one complex root. Thus, the quadratic equation has two complex roots when b 2 - 4ac < 0. Nature of Roots When D < 0Īnd it gives us two complex roots (which are different) as the square root of a negative number is a complex number. Thus, the quadratic equation has two real and different roots when b 2 - 4ac > 0. Nature of Roots When D > 0Īnd it gives us two real and different roots. Since the discriminant D is in the square root, we can determine the nature of the roots depending on whether D is positive, negative, or zero. So this can be written as x = (-b ± √ D )/2a. The roots of quadratic equation formula is x = (-b ± √ (b 2 - 4ac) )/2a. The discriminant of the quadratic equation ax 2 + bx + c = 0 is D = b 2 - 4ac. We can determine the nature of the roots by using the discriminant. But for finding the nature of the roots, we don't actually need to solve the equation. two real and equal roots (it means only one real root)įor example, in the above example, the roots of the quadratic equation x 2 - 7x + 10 = 0 are x = 2 and x = 5, where both 2 and 5 are two different real numbers, and so we can say that the equation has two real and different roots.The nature of the roots of a quadratic equation talks about "how many roots the equation has?" and "what type of roots the equation has?". So the best methods that always work for finding the roots are the quadratic root formula and completing the square methods. Note that the factoring method works only when the quadratic equation is factorable and we cannot find the complex roots of the quadratic equation using the graphing method. We can observe that the roots of the quadratic equation x 2 - 7x + 10 = 0 are x = 2 and x = 5 in each of the methods. Therefore, the roots of the quadratic equation are x = 2 and x = 5. Solution: To solve this, we just need to graph f(x) = x 2 - 7x + 10 and identify the x-intercepts.
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