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

Here, 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.

2. The standard form of a quadratic equation is

ax2 + bx + c  =  0

3. There are three methods to find the two zeros of a quadratic equations.

They are,

(i) Factoring

(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 in standard form ax2+ bx + c = 0 is -b/a.

6. The product of the zeros of the quadratic equation in standard form ax2+ bx + c = 0 is c/a.

7. If two zeros of a quadratic equation ax2+ bx + c = 0 are reciprocal to each other, then their product is 1 or

c  =  a

8. If two zeros of a quadratic equation ax2+ bx + c = 0 are equal in magnitude, but opposite in sign, then their sum is equal to zero or

b  =  0

9. If we know the two zeros  of a quadratic equation, the formula given below can be used 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.

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