 # What is the relationship between the coordinates of the image of the quadratic function y = AX2 + BX + C and the intersection of the x-axis and the root of the univariate quadratic equation AX2 + BX + C = 0?

## What is the relationship between the coordinates of the image of the quadratic function y = AX2 + BX + C and the intersection of the x-axis and the root of the univariate quadratic equation AX2 + BX + C = 0?

The intersection with the X axis is the root of the equation
If there are two intersections with the X axis, there are two roots
No intersection, no root

### If the univariate quadratic equation AX2 + BX + C = 0 has two real roots X1 and X2, then for the quadratic function y = AX2 + BX + C, you can explore one that only uses x1, Do x2 and a represent another analytical expression of this function? Find out this analytical formula, give an example of a quadratic function, and write its analytical formulas in different forms

It can be written as: y = a (x-x1) (x-x2)
For example, y = 2x ^ 2-8x + 6 can be written as y = 2 (x-1) (x-3)

### It is known that the univariate quadratic equation x-4x + B = 0 has two equal real roots, and the image of the quadratic function y = x-bx + C passes through a (1, Y1) B (3.y2) C (2.7, Y3) find the size relationship of Y1, Y2 and Y3 Please answer as soon as possible! The sooner the better. There's no point if it's late!

x ²- 4X + B = 0 has equal real roots
b=4
x ²- 4X + C = 0 symmetric about x = 2
y1=y2>y3

### Given the polynomial ax squared - BX + C, when x = 1, its value is 0, and when x = - 2, its value is 1, find one of the univariate quadratic equation AX squared + BX + C = 0 with respect to X sit back and wait. A root of

Obviously - 1 is a root of the equation you said, A-B + C = 0. From this, we can get that a root of the univariate quadratic equation AX square + BX + C = 0 about X is - 1. If it is difficult to understand, the specific instructions are as follows: 4A + 2B + C = 1 is easy to obtain, B = 1 / 3-A, C = 1 / 3-2a, so the univariate quadratic equation AX square + BX + C = 0 about X can be written

### The image with known quadratic function y = ax ^ 2 + bx-3 passes through point a (2,3) B (- 1,0) to find the analytical formula of quadratic function

∵ quadratic function y = ax ²+ The image of bx-3 passes through point a (2,3) B (- 1,0)
∴4a＋2b－3＝3 a－b－3＝0 ∴a＝2 b＝﹣1
The analytical formula of quadratic function is: y = 2x ²－ x－3

x ²- y ²
=（x+y）（x-y）
=-4x6
=-24