Math If m,n satisfies the equation x 2 m x 15 x 3 x n find the value of m n

Let's start with the first question.

Give both sides of the equal sign to x 2 + mx-15 = x 2 + (n + 3) x + 3n and then compare the lines.

It is concluded that m=n+3 3n=-15

Find m and n

We get m=-2 n=-5

The result is m+n=-7

Of course, here you can also understand that the equations on both sides of the equal sign are merged with similar terms.

We get (m-n-3)x+(-15-3n)=0 because this is a constant true problem, so no matter what value x takes, it is true.

So m-n-3=0 and -15-3n=0

Then ask for it and get it.

What about the second question. Let's talk about the formula merge first, and become (x+2) 2 + y-3) 2=0, which is mostly the case, and it will become the addition of two perfectly square subs, because the perfect square number is greater than or equal to zero.

So, the equation only holds when x+2=0 and y-3=0.

So x=-2, y=3

So x y=1 9

What about the third question. The first question, with its discriminant formula, simplification =(m-2) 2+4 is the formula is greater than zero, so the first question is proved.

The second question is that since it is an opposite number, the sum of its two roots is 0, and Vedic theorem is -(m+2)=0

So m=-2

Then we can take in the original equation and get x 2 - 3=0, and we can find out whether its solution is root number 3 or negative root number 3, and the idea is like this.

1. (x+3)(x+n)=x 2+(3+n)x+3n=x 2+mx-15, then 3+n=m, 3n= -15, so n= -5, m= -2, m+n= -7

2, x 2+y 2-4x+6y+13=(x-2) 2+(y+3) 2=0, then x=2, y= -3, then x y= 2 (-3)= 1 8

1) = (m+2) 2 - 4(2m-1)=m 2 - 4m +8=(m-2) 2+4 > 0, i.e., the equation has two real roots of inequalities, regardless of the value of the real number m.

2) The two roots of the equation are opposite to each other, then x1+x2=0, that is, m+2=0, m= -2, then the equation is x2=5, then x= 5

1)x^2+mx-15=(x+3)(x+n) ==>x^2+mx-15=x^2+3x+nx+3n 3n=-15 m=3+n n=-5,m=-2

2) x 2+y 2-4x+6y+13=0 ===>(x-2) 2+(y+3) 2=0 x=2 and y=-3 x y=1 8

3) Discriminant = (m+2)2-4(2m-1) = (m-2)2+4>0 So no matter what value the real number m takes, the equation has two real roots of inequalities.

x1+x2=0 m+2=0 m=-2 x= root number 5

1.Multiply the right side and correspond to the coefficients of x on both sides one by one, and solve it to get n=-5, m=-2, m+n=-7

2.Seeing this kind of problem, the recipe is: the original formula = (x-2) 2+(y+3) 2=0, because the sum of squares is always greater than or equal to 0, so x=2, y=-3, x y=2 -3=

1.As long as it is proved that (m+2) 2-4(2m-1)>0, the formula is disassembled to obtain m 2-4m+8=(m+2) 2+4, because (m+2) 2 is greater than or equal to 0, and after adding 4, it must be greater than 0, so no matter what value m takes...

2, is the opposite number, the sum of the two roots is 0, according to Veda's law, —(m+2)=0, so m=-2, the original formula =x 2-5=0, so x=positive and negative root number five.

1. From x 2 + mx-15 = (x + 3) (x + n), we get: the right side of the equation is equal to x 2 + (3 + n) x + 3n, then.

m=3+n,-15=3n, so n=-5,m=-122, the left side of the equation can be transformed as: (x-2) 2+(y+3) 2=0, so x=2, y=-3, x y=2 (-3)=1 8

3. (1) The use of the method of verification can be;

2) Using Vedr's theorem: x1+x2=-(m+2)=0, m=-2 is obtained, which can be solved by substituting the original equation.

2m-n)x＞5n-m

From the form of the solution set, we can know that 2m-n 0, that is, there is no 2m n and (5n-m) (2m-n) = 10 7

i.e. 7 (5n-m) = 10 (2m-n).

Get n = 3m 5 m, get m 0, n 0

mx+n＜0

mx -n, because m 0, so.

1 m)mx (1 xiangna m)(-n)=-n m=-3 5 gives x -3 5

When x tends to 1, the denominator tends to 0

The limit value is a non-zero constant.

Therefore, the numerator tends to 0

x^2+mx+n=1+m+n=0

And you can reduce x-1 to get x+n=5

i.e. n = 5 - 1 = 4 and m = -5

(1)x^2+mx-15=(x+3)(x+n)==>x^2+mx-15=x^2+3x+nx+3n3n=-15

m=3+nn=-5,m=-2

2) x 2+y 2-4x+6y+13=0===>(x-2) 2+(y+3) 2=0x=2 and.

y=-3x^y=1/8

3) Discriminant = (m+2)2-4(2m-1) = (m-2)2+4>0 So no matter what value the real number m takes, the equation has two real roots of inequality x1+x2=0

m+2=0m=-2

x= root number 5

1. (x+3)(x+n)=x 2+(3+n)x+3n=x 2+mx-15, then 3+n=m,3n=

15, so n=

5，m=2，m+n=

72, x 2+y 2-4x+6y+13=(x-2) 2+(y+3) 2=0, then x=2, y=

3, then x y=

m+2)^2

4(2m-1)=m^2-4m

8=(m-2)^2+4

0, that is, no matter what value the real number m takes, the equation has two real roots of inequality 2) The two roots of the equation are opposite to each other, then x1+x2=0, that is, m+2=0, m=-2 At this time, the equation is x2=5, then x= 5

Let's start with the first question.

Give both sides of the equal sign to x 2 + mx-15 = x 2 + (n + 3) x + 3n and then compare the lines.

It is derived that m=n+3

3n=-15

Finding m and n yields m=-2

n=-5 results in.

m+n=-7

Of course, here you can also understand that the equations on both sides of the equal sign are combined to obtain (m-n-3)x+(-15-3n)=0, because this is a constant true problem, so no matter what value x takes, it is true, so m-n-3=0 and.

15-3n=0

Then ask for it and get it.

What about the second question. Let's talk about the formula merge first, and become (x+2) 2

y-3)^2=0

Most of these problems are like this, and it becomes the addition of two perfectly square subs, because the perfect square number is greater than or equal to zero.

So, the equation is only true if x+2=0 and y-3=0, so x=-2, y=3

So x y=1 9

What about the third question. The first question, with its discriminant formula, simplification =(m-2) 2+4 is the formula is greater than zero, so the first question is proved.

The second question is that because it is an opposite number, the sum of its two roots is 0 according to Veda's theorem.

m+2)=0

So m=-2

Then bring in the original formula.

You can get it.

x 2-3=0 to find its solution is.

Root number 3 is yes.

The negative root number 3 idea is just that.

(mx-n)(x-2)/(x-1)=0

The three roots are n m, 2, 1

m>n>0, know n m<1<2

It can be known from the needle lead method.

The solution set of inequalities is: n m<=x<=1 or x>=2

Analysis: Using the rule of multiplication of polynomials and polynomials, the left side of the equation is compared, and the problem is transformed into an equation about m,n to determine the value of m,n by comparing the coefficients of the corresponding terms on the left and right sides

Answer: Solution: (x+1)(x-3)=x2-2x-3=x2+mx+n,m=-2,n=-3

Therefore, choose B Comment: This question examines the polynomial multiplication polynomial, and the algorithm needs to be mastered, and the key to solving the problem is to use the equal coefficient of the corresponding term