What is the relationship between the root case of a quadratic equation and the value of 4ac in squar

Any unary quadratic equation ax 2 bx c 0(a≠0) can be configured as (x+(b 2a)) 2=b 2-4ac, because a≠0, from the meaning of the square root, the sign of b 2 4ac can determine the case of the root of the unary quadratic equation b 2 4ac is called the discriminant of the root of the unary quadratic equation ax 22 bx c 0(a≠0), which is denoted by " " (read delta), that is, b 2 4ac

1 Discrimination of the roots of the unary quadratic equation ax 2 bx c 0(a≠0) (1) When 0, the equation has two unequal real roots; (2) When 0, the equation has two equal real roots; (3) When 0, the equation has no real roots (1) and (2) combined: When 0, the equation has two real roots The above conclusion is also true in turn It can be specifically expressed as: In the unary quadratic equation ax 2 bx c 0(a≠0), when the equation has two unequal real roots, 0;When the equation has two equal real roots, 0;When the equation has no real roots, 0.

Note that the discriminant formula for the root is =b2 4ac, not =sqrt(b24ac). sqrt refers to the root number) Formula for finding the root of a quadratic equation: When δ=b 2-4ac 0, x=[-b (b 2-4ac) (1 2)] 2a When δ=b 2-4ac 0, x= 2a (i is an imaginary unit) Quadratic equation matching method:

ax 2+bx+c=0(a,b,c is constant) x 2+bx a+c a=0 (x+b 2a) 2=(b 2-4ac) 4a 2 x+b 2a= (b 2-4ac) (1 2) 2a x=[-b (b 2-4ac) (1 2)] 2a

Let's let *=b2-4ac, then there is: 1. If * 0, then the original equation has two unequal real roots; 2。

If * 0, then the original equation has no real roots, and there is a pair of conjugate imaginary roots; 3。If *=0, then the original equation has two equal real roots, i.e., there is a unique solution. Hope.

b 0 5-4ac>0 has two different real roots, b 0 5-4ac=0 has two identical real roots, b 0 5-4ac<0 has no real roots.

Equal to 0 has one real root, greater than 0 has two unequal real roots, less than 0 has no real root.

The unary quadratic equation ax 2+bx+c=0

Left side of the formula = a(x+b 2a) 2+c-b 2 4a=a(x+b 2a) 2+(4ac-b 2) 4a=0

i.e. (x+b 2a) 2=-(4ac-b 2) 4a 2=(b 2-4ac) 4a 2,a≠0,4a 2 0,(x+b 2a) 2 0

When b 2-4ac < 0, contradictory. No solution.

When b 2-4ac 0, the equation can be solved, so the positive and negative of b 2-4ac determines whether the unary quadratic equation has a solution, which is called the discriminant formula of the root.

First of all, there is an arbitrary quadratic equation.

ax²+bx+c=0 a≠0

a(x² +b/a x)+c=0

a(x² +2*b/2a x)=-c

a[x +2*b 2a x +(b 2a) b 2a) ]ca[x +2*b 2a x +(b 2a) ]b 4a=-ca[x +2*b 2a x +(b 2a) ]b 4a -ca(x+b 2a) =b -4ac) 4a(x+b 2a) =b -4ac) 4a on both sides of the root number.

We find that the denominator 4a on the right must be greater than 0, and the numerator may be less than 0, that is, the discriminant formula is b -4ac

If this value < 0, there is no solution to the equation; =0, there are two identical real solutions (generally not called 1 solutions); >0, there are two solutions.

Using the discriminant formula of the root of a quadratic equation ( =b -4ac) to determine the root of an equation The root of the unary quadratic equation ax +bx+c=0(a≠0) has the following relationship with =b -4ac:

When 0, the equation has two unequal roots of two real numbers;

When =0, the equation has two equal roots of two real numbers;

When 0, the equation has no real roots

The above conclusion holds true in reverse

Example: The most accurate case for the unary quadratic equation x (2m 1) x m 2 0 real roots of x is ( ).

a. There are real roots.

b. No solid roots.

c. There are two equal real roots.

d. There are two unequal solid roots.

Solution: δ2m 1)] 4 1 ( m 2) 4m 4m 1 4m 8

4m 9 0, the equation has two unequal real roots, so choose: d

b 2-4 ac is greater than zero, so the equation has two unequal real roots, b 2-4 ac=0, so the equation has two equal real roots, b 2-4 ac is less than zero, so the equation has no roots.

An equation is an equation, and a quadratic equation is one in which the highest order term of the equation is 2 and there is an unknown.

The unary quadratic Shen Changcheng is: ax 2+bx+c=0 shift: ax 2+bx=-c

Multiply the two sides by 4a: 4(ax) 2+4abx=-4ac plus b 2: 4(ax) 2+4abx+b 2=b 2-4ac to inflated the perfect flat way:

2ax+b) 2=b 2-4ac From here, it can be seen that the poor filial piety is old, only when b 2-4ac>=0 x will have a solution, if b 2-4ac<0 will definitely not be solved.

b 2a is the vertex abscissa of the unary quadratic function image, which is: y=ax 2+bx+c

y=a(x^2+b/ax)+c

a(x+b 2a) 2-(b 2 4a)+c, it can be seen that when x = b 2a, y obtains the maximum value (a<0) or the minimum value (a>0).

The binary one-dimensional equation of b's squared -4ac=0 can be reduced to a perfect square.

Because the equation has two equal real roots, it can be read into a perfect flat way.

If it is less than 0, the equation has no solution, and if you are learning a virtual function, there is an i outside the root number of the root formula.

There is no root in the range of real numbers, and the formula for finding the root in the range of complex numbers is x=[-b i (b 2-4ac)] 2a

Derivation process:

The unary quadratic equation is: ax 2+bx+c=0

Shift item: ax 2+bx=-c

Multiply the two sides by 4a: 4(ax) 2+4abx=-4ac plus b 2: 4(ax) 2+4abx+b 2=b 2-4ac to make a perfect flat:

2ax+b) 2=b 2-4ac, only when b 2-4ac>=0 x will have a solution, if b 2-4ac<0 can not be solved.

Therefore, b 2-4ac is discriminant.

I can push it myself, but there is one in the book

The unary quadratic equation is: ax 2+bx+c=0

Shift item: ax 2+bx=-c

Multiply both sides by 4a: 4(ax) 2+4abx=-4ac

Add b 2: 4(ax) 2+4abx+b 2=b 2-4ac

Flattened perfectly: (2ax+b) 2=b 2-4ac

From this, it can be seen that x will only have a solution if b 2-4ac>=0, and if b 2-4ac<0 will definitely not be solved.

b 2a is the vertex abscissa of the unary quadratic function image, which is: y=ax 2+bx+c

y=a(x^2+b/ax)+c

a(x+b/2a)^2-(b^2/4a)+c

It can be seen that when x = -b 2a, y obtains a maximum value (a<0) or a minimum value (a>0).

Isn't there an ominous explanation in the textbook?

For a quadratic equation, a·x +b·x+c=0, where a≠0, the positive and negative representations of the discriminant formula can be derived by the matching method

a·x²+b·x+c＝0

a·(x+b/2a)²+c-b²/4a＝0

x+b/2a)²=(b²-4ac)/4a²

So when b -4ac 0, the unary quadratic equation a·x +b·x+c=0 has two different real roots;

When b -4ac 0, the unary quadratic equation a·x +b·x+c=0 has two identical real roots;

When b -4ac 0, the unary quadratic equation a·x +b·x+c=0 has no real root.

For -2a b, we can see that x+b 2a 0 is the axis of symmetry of this unary quadratic function image.

When b -4ac 0 (x+b 2a) = (b -4ac) 4a

x+b/2a=±√(b²-4ac)/2a

The general formula y=ax 2+bx+c of a quadratic function is converted into a vertex formula.

y=ax^2+bx+c

a(x^2+bx/a+c/a)

a(x 2+bx a+b 2 4a 2)-b 2 4a+c=a(x+b 2a) 2-b 2 4a+c=a(x+b 2a) 2-(b 2-4ac) 4a-b 2a represents the axis of symmetry.

The dots (-b 2a ,(b 2-4ac) 4a ) represent vertices.

ax^2+bx+c=0 a[x^2+b/ax+(b/2a)^2]+c-a*(b/2a)^2=0

a(x+b/2a)^2=b^2/4a-c

x+b/2a)^2=(b^2-4ac)/4a^2∵(x+b/2a)^2≥0，4a^2≥0

b 2-4ac<0, the equation has no real root.

b 2-4ac 0, x+b 2a = [under root number (b 2-4ac)] 2a

i.e. x=-b 2a [under the root number (b 2-4ac)] 2a

It works. There is no time when the term is b=0

x 2-9=0 is x 2+0x-9=0

a=1 b=0 c=-9

b^2-4ac=36

x=(-b±√⊿2a)=±3

According to the matching method, the equation can be matched as: the left wingback is completely flat, the right side is δ if δ< 0, then the square number is less than zero, so there is no real root.

ax²+bx+c=0

a(x+b/2a)²=b²-4ac

ax^2+bx+c=0

a(x^2+bx/a+c/a)=0

a[x^2+bx/c+b^2/(4a^2)-b^2/(4a^2)+c/a]=0

a[x+b (2a)] 2-(b 2-4ac) (4a)]=0x+b (2a)] 2=(b 2-4ac) (4a 2) So, when b 2-4ac 0, the equation has a real solution.

The Vedic formula came about to understand the quadratic equations, which can certainly be proven.