A. 1
Tip: There are four points to be paid attention to in the definition of quadratic equation with one variable: it contains an unknown number; The maximum number of unknowns is 2; The coefficient of quadratic term is not 0; It is an integral equation, in which equation ① does not pay attention to the condition of missing m≠3; Equation 2 mistakenly regards k as an unknown number; The left side of equation (4) is not an algebraic expression.
A: A.
2. The quadratic coefficient of 2.x = x2 is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Tip: By removing brackets, moving terms and merging similar terms, the equation can be transformed into a general form. When determining the coefficients of quadratic term, linear term and constant term, the original equation should be transformed into a general form first, and symbols should not be omitted at the same time.
Answer: 0
3. Transform the equation (2x+ 1)(x-3)=x2+ 1 into the general form of _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Hint: The equation in the problem is not in a general form. We should first remove the brackets, move the terms, and merge the similar terms to turn the equation into a general form. In the process of conversion, the symbol of the moved item should be changed and similar items should be merged accurately.
Answer: x2-2x-4=0 -5.
4. If the constant term of Equation 4(x+m)2+2m-2=0 of X is 0, then the value of M is _ _ _ _ _ _ _ _.
Tip: Turn the equation into a general form and list the equation about m according to the constant term 0.
Answer: or-1
5. If the equation ax2-(a+ 1)x-3=0 is a quadratic equation, then the value of a is _ _ _ _ _ _ _.
It is suggested that to determine the range of a, we should first convert the original equation into a general form, and then calculate the range of a according to the fact that the coefficient of quadratic term can not be zero.
Answer: a≠0
A school plans to increase the number of high school students from 3000 to 3630 in two years. If the average annual growth percentage is x, the equation listed is ().
A.( 1+x)2 = 3 630 b . 3 000( 1+x)= 3 630
C.3 000( 1+x)2=3 630
Tip: If the average annual growth percentage is x, it will increase to 3 000( 1+x) in the first year and 3 000( 1+x)2 in the second year.
Answer: c
7. If the sum of quadratic coefficient, linear coefficient and constant term of quadratic equation 2x2+(k+8)x-(2k-3)=0 is 5, then k = _ _ _ _ _ _ _ _ _ _
Tip: According to the general form of quadratic equation with one variable, it is the key to solve this problem to write out the quadratic term coefficient, the linear term coefficient and the constant term correctly.
Solution: ∫2+(k+8)+[-(2k-3)]= 5, ∴k=8.
Answer: 8
8. The coefficient of the first term of the unary quadratic equation x2=2 is _ _ _ _ _ _ _.
Tip: If the quadratic equation of one variable is transformed into a general form, the shift term is x2-2=0.
Answer: 0
9. The equation about x (k+3) (k-1) x2+(k-1) x-5 = 0 is known.
(1) When k is taken, is this equation linear? And find the solution of this equation;
(2) When k is taken, is this equation a quadratic equation? And write the quadratic term coefficient, linear term coefficient and constant term of this unary quadratic equation.
Hint: (1) If the original equation is linear, it must satisfy (k+3)(k- 1)=0 and k- 1≠0.
The solution is k=-3.
(2) If the original equation is a quadratic equation, it must satisfy (k+3)(k- 1)≠0, that is, k≠-3, k≠ 1. Its quadratic coefficient is (k+3)(k- 1.
Comprehensive? Apply? reform
10. It is known that the equation about x (m-3) -x=5 is a quadratic equation. Find the value of m.
Tip: Using the definition of quadratic equation with one variable, the key is to pay attention to the condition that the coefficient of quadratic term is not zero.
Solution: According to the meaning of the question, m2-7=2, m-3≠0,
So we can only take m=-3. That is, when m=-3, the equation (m-3) -x=5 is a quadratic equation.
1 1.a is a quadratic term coefficient, b is a linear term coefficient, and c is a constant term, which satisfies +(b-2)2+|a+b+c|=0, so we can solve the unary quadratic equation.
Hint: The key to this problem is to understand the meanings of arithmetic root, complete square number and absolute value, namely ≥0, (b-2)2≥0, |a+b+c|≥0. Only when all items are 0, the total is 0. This question examines the degree of mastery of the knowledge learned, which is closely related to the newly learned knowledge of quadratic equations in one variable.
Solution: from +(b-2)2+|a+b+c|=0,
The solutions are a = 1, b = 2 and c =-3.
∫a is a quadratic term coefficient, B is a linear term coefficient, and C is a constant term.
The equation is x2+2x-3=0.
12. In order to reduce the burden on students, a printing factory plans to reduce the book fee for students by 36% within two years. If the average annual reduction percentage is x, what is the equation about x?
Tip: To understand the cost reduction of 36%, we can set the original cost unit as 1, which will be 64% after two years.
Answer: (1-x)2=64%.
13. Fold a 22 cm long iron wire into a rectangle with an area of 30 cm 2. What is the length and width of this rectangle?
Tip: Use the rectangular area formula to list the corresponding equations, and use the list method to find the length and width of the rectangle.
Solution: Let the length of a rectangle be x cm and the width be (-x) cm. According to the meaning of the question, x( -x)=30, that is
x2- 1 1x+30=0。
List:
x 1 2 3 4 5 6
x2- 1 1x+30 20 12 6 2 0 0
When the length of the rectangle is 5 cm and the width is 1 1-x=6 cm (irrelevant); When the length of the rectangle is 6 cm, the width is 1 1-6=5 cm.
Answer: The length of a rectangle is 6 cm and the width is 5 cm.
Review? Warm up? prospect
14. (Tianshui Senior High School Entrance Examination in Gansu Province in 2005) In June this year, the industrial output value of 5438+0 reached 500 million yuan, and the total output value in the first quarter was1800 million yuan. If the average growth rate in the next two months is x, the equation that can be listed according to the meaning of the question is _ _ _ _ _ _ _. (please put it.
Tip: The first quarter is the output value of three months, and 5+5 (1+x)+5 (1+x) 2 =18.
Answer: 5x2+ 15x-3=0.
15. (classical playback) The following two groups of equations are all quadratic equations about x ().
a . x2 = 0.5x 2-5 = 0 b . MX+m2 = 0.3x 2-5 = 0
C.(m- 1)x2+4x=0，ax2+bx+c=0 D. +x2=0，=0
Hint: Both terms in A are quadratic equations with one variable; In item B, the first one is a univariate linear equation about X and the last one is a univariate quadratic equation. The first item in item C is missing, and the last item is missing; Neither is in item D.
A: A.
16. (classic playback) A store bought a batch of daily necessities at the original price of 16 yuan. After selling for a while, in order to get more profits, the store decided to raise the price. Through experiments, it is found that if you sell at the price of each piece in 20 yuan, you can sell 360 pieces per month, and if you sell at the price of each piece in 25 yuan, you can sell 2 10 pieces per month.
(1) Try to find the relationship between y and x;
(2) When the price is X yuan, what is the profit of the goods at this time?
Prompt: the relationship between (1)Y and x can be set as y = kx+B. Let x=20, Y = 360X=25, y=2 10, and substitute them to find the values of k and b respectively. (2) Commodity profit = (commodity price-commodity purchase price) × commodity sales quantity.
Solution: (1) let the relationship between y and x be y=kx+b,
X=20, y = 360X=25 and y=2 10 are substituted into y=kx+b respectively, and we can get
The solution is k=-30 and b=960.
The relationship between y and x is y=-30x+960.
(2) When the selling price is X yuan, the profit of the commodity is (x- 16)(-30x+960).