Quadratic equations Vedic theorem .

By the first equation, there is:

x1+x2=-p...1)

x1*x2=q...2)

By the second equation, there is:

x1+x2+2=-q...3)

x1+1)*(x2+1)=1+(x1+x2)+x1*x2=p...4)

Bring (1) into (3) and get:

p+2=-q...5)

Bring (2) into (4) and get:

1-p+q=p...6)

Finally, to solve this system of binary equations, we get:

p=-1;q=-3

All questions are done.

By the Vedic theorem.

Get x1+x2=-p, x1x2=q

x1+x2+2=-q,(x1+1)(x2+1)=p, substituting the upper two formulas into the lower two formulas.

This results in p-q=2 and 2p-q=1

p=-1,q=-3

x1+x2=-p

x1*x2=q

x1+1+x2+1=-q

x1+1)(x2+1)=p

Solve a system of quaternary linear equations. Can.

Vedder's theorem explains the relationship between roots and coefficients in a quadratic equation.

The French mathematician François Veda established the relationship between the roots of equations and the coefficients in his book "On the Identification and Revision of Equations" and proposed this theorem. Because Veda first developed this relationship between the roots and the coefficients of modern number equations, people call this relationship Veda's theorem.

Da's theorem plays a unique role in finding the function of the root of the liquid, discussing the symbols of the roots of quadratic equations, solving symmetric equations, and solving some problems about quadratic curves.

The discriminant formula of the root is a sufficient and necessary condition for determining whether an equation has a real root, and Veda's theorem explains the relationship between the root and the coefficient. Regardless of whether the equation has a real root or not, the relationship between the root and the coefficient of the real coefficient one-element quadratic spine equation is suitable for Vedic theorem sail guess. The combination of discriminant formula and Vedic theorem can more effectively explain and determine the condition and characteristics of the root of a quadratic equation.

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The Vedic Theorem is a theorem that explains the relationship between roots and coefficients in a quadratic equation, and was proposed by François Lauda Veda.

1. The significance of Veda's theorem.

Vedda's theorem plays a unique role in finding the symmetry function of the root, discussing the sign of the root of a quadratic equation, solving the system of symmetric equations, and solving the problem of quadratic curves. The most important contribution of this theorem is the advancement of algebraic coarseness, and the first systematic introduction of algebraic notation has advanced the development of equation theory.

2. François Veda.

François Veda was born in Poitou, France, in 1540. He died in Paris on the 13th of December 1603. When he was young, he studied law and became a lawyer, and later engaged in political activities, became a member of parliament, and deciphered the code of the enemy army during the war against Spain.

Veda devoted his life to the study of mathematics and was the first to consciously and systematically use letters to represent known numbers, unknowns, and their powers, contributing to significant advances in the study of algebraic theory.

III. The main achievements of Veda.

Veda's most important contribution was the advancement of algebra, and he was the first to systematically introduce algebraic notation to advance the development of equation theory. Veda uses the word "analysis" to summarize the content and methods of the current era.

He created a large number of algebraic symbols, replaced unknown numbers with letters, and systematically elaborated and improved them.

The solution of the third and fourth order equations points out the relationship between the root and the coefficient. The trigonometric solution of the irreducible case of the cubic equation is given. He is the author of many books, such as "Introduction to Analytical Methods" and "On the Identification and Revision of Equations".

The Vedic theorem of the unary cubic equation is:

Let the cubic equation be ax 3 + bx 2 + cx + d = 0.

The roots of the three bends are x1, x2, and x3, respectively, and the equation can be expressed as a(x-x1)(x-x2)(x-x3)=0.

That is, ax 3-a(x1+x2+x3)x 2+a(x1*x2+x2*x3+x3*x1)-ax1*x2*x3=0.

Comparing the original equation ax 3+bx 2+cx+d=0, we can see that:

x1+x2+x3=-b/a。

x1*x2+x2*x3+x3*x1=c/a。

x1*x2*x3=-d/a。

Theorem significance

Vedadin is a clever attack on the symmetry function of the root, and the discussion of the symbols of the roots of quadratic equations, the solution of symmetric equations, and the solution of some problems related to quadratic curves are all highlighted.

The discriminant formula of the root of a quadratic equation is (a, b, c are the quadratic coefficients, the coefficient of the primary term, and the constant term of the quadratic equation, respectively), and the relationship between Vedica's theorem and the discriminant formula of the root is even more inseparable.

The discriminant formula of the root is a sufficient and necessary condition for determining whether an equation has a real root, and Veda's theorem explains the relationship between the root and the coefficient. Regardless of whether the equation has a real root or not, the relationship between the root and the coefficient of a quadratic equation with real coefficients fits Vedica's theorem; The combination of discriminant formula and Vedic theorem can more effectively explain and determine the condition and characteristics of the root of a quadratic equation.

The x1 times x2 formula Vedic theorem is a quadratic equation of one element. That is, ax plus bx plus c is equal to 0, a is not equal to 0 and is equal to b degree 2 minus 4ac is greater than or equal to 0 if the two roots are x1 and x2, then x1 plus x2 is equal to negative b divided by a, x1 times x2 is equal to c divided by a, and there is only one unknown unary, and the highest number of unknowns is 2 quadratic integer equations are called unary quadratic equations.

x1 times x2 formula: Features of Veda's theoremThe sum of the two roots of the equation is equal to the coefficient of the first term divided by the opposite of the coefficient of the quadratic term, and the product of the two roots is equal to the constant term divided by the coefficient of the quadratic term, Vedda's theorem illustrates the relationship between the roots and the coefficients in the quadratic equation, and the Vedic theorem shows its unique role in finding the symmetry function of the root, discussing the coarse-sign solution of the root of the quadratic equation to the system of symmetry equations, and solving some problems related to the quadratic curve.

Regardless of whether the equation has a real root or not, the relationship between the root and the coefficient of the real coefficient quadratic equation is suitable for Vedder's theorem, and the combination of the discriminant formula and Vedder's theorem is more effective in explaining and judging the condition and characteristics of the root of the one-element quadratic closed equation.

Veda's theorem proves the relationship between roots and coefficients in a univariate nth order equation.

Contents of the Vedic theorem of a quadratic equation:

In the unary quadratic equation ax 2+bx+c=0 a≠0, the two x1 and x2 have the following relationship:

x1+x2=-b/a,x1×x2=c/a

The two roots of the equation are related to each number in the equation as follows: x1+x2= -b a, x1·x2=c a (also known as Vedda's theorem).

When the two roots of the equation are x1 and x2, the equation is: x2-(x1+x2)x+x1x2=0 (inversely derived from Veda's theorem).

√(a+4)²+b-1|=0

a+4|+|b-1|=0

So a=-4, b=1

So it's kx -4x+1=0

If there are two unequal real roots, the discriminant is greater than 0

16-4k>0

k<4 and this is a quadratic equation.

So the x-coefficient k≠0

So k<4 and k≠0

a(x-x1)(x-x2)=ax 2-a(x1+x2)x+ax1x2 and a(x-x1)(x-x2)=ax 2+bx+c, so ax 2-a(x1+x2)x+ax1x2=x 2+bx+cThe coefficients of x 2 should be equal (a=a), the coefficients of x should be equal (-a(x1+x2)=b), and the constant term coefficients should be equal (ax1x2=c).

x1x2=c/a

The root finding formula is used for calculation and derivation.

Let the simplified quadratic equation be: x 2 + px + q = 0

Then: x1=[-p+ (p -4q)] 2 x2=[-p- (p -4q)] 2

x1+x2=-p/2+√(p²-4q)/2+[-p/2-√(p²-4q)/2]

px1*x2=(-p/2)²-p²-4q)/4=p²/4-p²/4+4q/4

q is expressed in words: the sum of the two roots of the simplified quadratic equation is equal to the opposite of the coefficient of the primary term, and the product of the two roots is equal to the constant term.

Suppose that the quadratic equation is: ax +bx+c=0(a≠0) and its two roots are: x1 and x2

Then, the unary quadratic equation can be written as: (x-x1)(x-x2)=0 Then, the formula: x -(x1+x2)x+x1x2=0 Since the formula is the original equation is the same unary quadratic equation, write the equation as:

x +(b a)x+c a=0 due to =

So: x1+x2=-b a

x1x2=c/a

Veda's theorem was proven.

For the two x1 and x2 of the general formula ax 2 + bx + c = 0 (a is not equal to 0), the original equation is always satisfied, so x-x1 and x-x2 are both the factors of the three terms on the left side of the original equation, so the original equation must be rewritten as a(x-x1) (x-x2) = 0 and the coefficient can be compared.