Original link: https://naichaduck.github.io/post/cc261ac0.html

## Mathematics from Beginner to Master

### foreword

I wanted to learn mathematics well, but found that I may not remember the knowledge of junior high school, so I found a “Complete Works of High School Mathematics Fundamentals and Solutions” on station B again.

### Math Learning Path

1 |
graph LR |

### cross multiplication

This is the basis for the transition between junior high school and high school.

$y=ax^2+bx+c$

where $a,b,c$ must all be integers. (i.e. no score)

example

$y=x^2-3x+2$

untie

1 -1

1 -2

The two 1s in the first column are from $X^2$ ($x^2=x *x$) The 1 and 2 in the second column are from $c$, in this question $c=2$, 2 can be split into $1* 2$ or $-1*-2$

The logic of splitting: make $b$ by splitting $c$ into the form of $c1*c2$

See below for details

After the cross is connected, the next step is to multiply the 1 and -2 of the first row to get $1*-2=-2$

Multiply the 1 in the second row by -1 to get $1*-1=-1$

Then add the two results $-2+(-1)=-3=b$

$b$ has been figured out. At this step, the cross multiplication is right. Finally, the simplified formula can be written horizontally and the 1 in the first column of the formula can be multiplied by x again.

1 plus -1 in the first row times 1 plus -2 in the second row

$(1 *x)+-1* (1 *x)+-2=(x-1)* (x-2)$

Finish

### Properties of Quadratic Functions

$y=ax^2+bx+c$

On the coordinate axis $y=ax^2+bx+c$ is a U-shaped curve, either the opening is facing up or the opening is facing down.

The ± of $a$ determines the opening direction + upward – downward.

$\Delta=b^2-4ac$

$\Delta$ is the position of the midpoint of the curve on the Y axis

The $\Delta>0$ function has two points on the X axis (y=0) $x_1,x_2$

The $\Delta=0$ function has a point on the X axis (y=0) $x_1=x_2$

The $\Delta<0$ function has no point on the X axis (y=0) and does not intersect the X axis

Where did $\Delta$ come from?

$b^2-4ac$ is derived from the general formula $ax^2+bx+c=0$:

The specific derivation process of $b^2-4ac$:

$ax^2+bx+c=0(a≠0)$

Divide both sides by a

get $x^2+b/aX+c/a=0$

Re-formulation is $x^2+b/aX+(b/2a)^2=-c/a+(b/2a)^2$

$x+b/2a)^2=b²-4ac/4a^2$

If $b²-4ac$ is greater than or equal to 0

$x=-b$±$b^2-4ac/2a$ under the square root

The symmetry axis of the function curve is $x=-\frac{b}{2a}$

### Unary quadratic function universal formula

It’s called Mitternacht Formel in German

$x_1,_2=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

### Veda’s Theorem

Use it when it’s hard to come by.

Premise: $\Delta>0$ (with $x_1,x_2$)

$x_1+x_2=-\frac{b}{a}$

$x_1x_2=\frac{c}{a}$

### Quadratic Function Inequality

Graphical judgment to see where the opening is.

The ± of $a$ determines the opening direction + upward – downward.

Exercise questions and answers related to the first lesson above

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