Properties of quadratic equations :
Here we are going to see the properties of quadratic equations which would be much useful to the students who practice problems on quadratic functions.
1. The zeros of a quadratic function f(x) = ax²+bx+c are nothing but the two values of "x" when f(x) = 0 or ax² + bx +c = 0.
"ax² + bx +c = 0" is called as quadratic equation
Finding the two zeros of a quadratic function or solving the quadratic equation are the same thing.
3. There are three methods to find the two zeros of a quadratic equations.
(ii) Quadratic formula
(iii) Completing square
4. If the two zeros of a quadratic equation are irrational, then the two zeros (roots) will occur in conjugate pairs. That is, if (m+√n)is a root, then (m-√n) is the other root of the same equation.
5. The sum of the zeros of the quadratic equation ax²+bx+c = 0 is -b/a
6. The product of the zeros of the quadratic equation ax²+bx+c = 0 is c/a
7. If one zero is reciprocal to the other root then their product c/a = 1 or c = a.
8. If one zero is equal to other root but opposite in sign then their -b/a = 0 or b = 0.
9. If we know the two roots of a quadratic equation, we can use the following formula to form the quadratic equation.
x² - (Sum of the roots)x + product of the roots = 0
10. The graph of any quadratic equation will be a parabola.
11. The zeros of a quadratic equation are the x-coordinates of the points where the parabola (graph of quadratic a function) cuts x-axis.
12. If the two zeros of a quadratic function are imaginary, then the graph (parabola) will never intersect x - axis.
13. The two x-intercepts of a parabola (graph of a quadratic function) are nothing but the zeros of the quadratic function.
14. x- coordinate of the vertex of the parabola is -b/2a and the vertex is [ -b/2a, f(-b/2a) ]
15. To know at where the parabola cuts y-axis or y-intercept of the parabola, we have to plug x = 0 in the given quadratic function.
16. f(x) = ax² + bx + c, if the sign of the first term (ax²) is negative, the parabola will be open downward. Otherwise, the parabola will be open downward.
17. The discriminant b² - 4ac discriminates the nature of the zeros of the quadratic equation ax² + bx + c = 0.
Let us see how this discriminant "b² - 4ac" can be used to know the nature of the roots of a quadratic equation.
After having gone through the stuff given above, we hope that the students would have understood "Properties of quadratic equations".
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