First, multiple choice questions

1. Of the following four statements, the correct one is ().

A. A quadratic equation with one variable has real roots;

B. The quadratic equation with one variable has real roots;

C. the quadratic equation of one variable has real roots;

D The unary quadratic equation x2+4x+5=a(a≥ 1) has real roots.

Answer d

2. A quadratic equation with one variable has two unequal real roots, then the following conditions are satisfied.

a.=0 b . >0

c.& lt0 d. ≥0

Answer b

3. (Meishan, XX, Sichuan) The two solutions of the known equation are respectively, and the value of is

A. d.3 in 7 BC

Answer d

4. One root of the equation x2+x–1= 0 is

A. 1- 0/–BC–1+d.

Answer d

5.(XX Shanghai) The quadratic equation x2+x ─ 1 = 0 is known, and the following judgment is correct ().

A. This equation has two equal real roots. This equation has two unequal real roots.

C. the equation has no real root d, and the root of the equation is uncertain.

Answer b

6.(XX Hubei Wuhan) If two equations =4, the value of is ().

a.8 b.4

c.2 d.0

Answer d

7.(XX Shandong Weifang) The quadratic equation x2-6x+2k=0 about X has two unequal real roots, so the range of the real number k is ().

a . k≤b . k & lt； c . k≥d . k & gt；

Answer b

8.(XX Chuxiong, Yunnan) The solution of quadratic equation x2-4=0 is ().

a.x 1=2，x2=-2 b.x=-2 c.x=2 d. x 1=2，x2=0

Answer a

9.(XX Kunming, Yunnan) The product of two roots of a quadratic equation is ()

a.- 1 b. -2 c. 1 d.2

Answer b

10.(XX Xiaogan, Hubei) The correct estimation of the equation is ().

a.b.

c.d.

Answer b

1 1.(XX Guilin, Guangxi) The solution of the quadratic equation in one variable is ().

a.，b，

c.，d，

Answer a

12. The solution of equation (x-5)(x-6)=x-5 is ().

A.x=5 b.x=5 or x=6 c.x=7 d.x=5 or x=7.

Answer d

Second, fill in the blanks

1. It is known that the univariate quadratic equation about x has real roots, so the range of m is.

answer

2. Given that x 1 and x2 are two real roots of equation x2+3x+ 1=0, then x12+8x2+20 = _ _ _ _ _ _ _.

Answer-1

3. Let x 1 and x2 be two roots of quadratic equation x2+4x-3=0.

2x 1(x22+5x2-3)+a =2, then a= ▲.

Answer 8

4. The solution of the unary quadratic equation is _ _ _ _ _ _ _ _ _ _ _.

answer

5. The solution of the equation is ▲.

answer

6.(XX Lianyungang, Jiangsu) If the equation x2-mx+3=0 about x has a real root, the value of m can be _ _ _ _ _ _ _. Just give a qualified value.

answer

7. If the equation ax2+2x+ 1=0 has two unequal real roots, the value range of the real number A is

Answer a

8. It is known that α and β are two real roots of the unary quadratic equation x2-4x-3=0, then the algebraic expression (α-3)(β-3)=.

Answer -6

9. If the real number m satisfies m2- m+ 1 = 0, then m4+m-4 =.

Answer 62

10. The two roots of the unary quadratic equation x2-5x+6=0 are X 1 and x2 respectively, so x 1+x2 is equal to.

a.5 b. 6 c. -5 d. -6

Answer a

1 1. The unary quadratic equation about x-x2+(2m+1) x+1-m2 = 0 has no real root, so the range of' value of m is _ _ _ _ _ _ _ _ _.

answer

12. It is known that the quadratic equation x2 +kx+1 =0 has two equal real roots.

Then k = ▲.

Answer 2

23. The root of the quadratic equation of X (x+3)(x- 1)=0 is _ _ _ _ _ _ _.

The answer is x= 1 or x=-3.

13. Write an unary quadratic equation with real roots _ _ _ _ _ _ _ _ _ _ _.

The answer is not unique, for example: x2-2x+ 1 =0.

14. The solution of the equation is.

answer

15. Reading materials:

If the two real roots of the unary quadratic equation ax2+bx+c=0(a≠0) are x 1 and x2, then the relationship between these two roots and the coefficient of the equation is as follows:

x 1+x2= -，x 1x2=

Fill in the blanks according to the above materials:

Given that x 1 and x2 are two real roots of the equation x2+4x+2=0, then+= _ _ _ _ _.

Answer -2

16. The sum of the two roots of the equation-1 is equal to.

Answer 2

Third, answer questions.

1.(XX Suzhou, Jiangsu) Solve the equation:.

answer

2.(XX, Guangzhou, Guangdong, 19, 10) It is known that the quadratic equation of one variable about X has two equal real roots, so evaluate it.

Because this equation has two equal real roots, so ⊿ =, the relationship between A and B can be obtained. Then, after simplification, A can be represented by an algebraic expression containing B, and the value of this score can be obtained.

Answer solution: ∵ There are two equal real roots,

= =, that is.

∵

∵ ,∴

3.(XX Qijiang County, Chongqing) Solve the equation: x2-2x- 1=0.

The answer to the equation: x2-2x- 1=0.

Solution:

∴ ;

4. (Bijie XX, Guizhou) It is known that there are two real numbers in a quadratic equation.

(1) Value range of real number;

(2) The value of when.

The answer is: (1),

Solve.

That is, the range of real numbers is.

(2) by.

If, that is, solution.

∵& gt； It doesn't matter. Give it up.

If so, it is known by (1).

So when ...

5.(XX Changzhou, Jiangsu) Solve the equation

answer

6.(XX Zhongshan, Guangdong) The quadratic equation of one variable is known.

(1) If the equation has two real roots, find the range of m;

(2) If the two real roots of the equation are 0 and +3 =3 respectively, find the value of m. ..

Answer: (1)δ=4-4m.

Because this equation has two real roots.

So 4-4m≥0, that is, m≤ 1.

(2) +=2 is obtained from the relationship between the root and the coefficient of a quadratic equation.

Plus sign +3 =3

So, =

Substitute = into the equation and get =

7.(XX Leshan, Sichuan) Choose one of the two questions A and B, do both questions, and only score with the A question.

Question A: Does a quadratic equation with one variable have a real root?

The range of (1) real number k;

(2) Assume and find out the minimum value of t 。

Question B: As shown in the figure (1 1), in the rectangular abcd, P is a point on the side of bc, which connects dp and extends, and the extension line of intersection ab is at point Q.

(1) If, the value of;

(2) If point P is any point on the edge of bc, verify it.

What I chose to do was _ _ _ _ _.

Answer question a

Solution: (1)∵ One-variable quadratic equation has real roots,

∴, ..................................................................................................................... 2 points.

That is to say,

Solution ................................................... 4 points.

(3) From the relationship between roots and coefficients, we get: …… 6 points.

Seven points.

∵ ,∴ ,

∴ ,

That is to say, the minimum value of t is -4. ................................................................................................................................................................

Theme b

(1) solution: the quadrilateral abcd is a rectangle,

Ab = cd, ab∨DC, ................................................. 1 min.

∴△dpc ∽△qpb, 3 points.

∴ ,

∴ ,

∴ ……………………………………………………………………………………………………………………………………………………………………………………………………………………………… 5 points.

(2) proof: by △dpc ∽△qpb,

, ... 6 points.

7 points

.......................... 10.

8. A quadratic equation of (xx Hubei Xiaogan) x,

(1) Find the range of p; (4 points)

(2) If the value. (6 points)

Answer: (1) Judging from the meaning of the question:

........................, two points.

Solution: 4 points.

(2) from,

........................ scored six points.

..........................., eight.

.........................., 9 points.

.............. 10 point

Description: 1. free

Substitute into the original evaluation formula to solve;

9. (Guangxi Fangchenggang XX Yulin) (6 points) When the real number k is what value, does the equation x -4x+3-k=0 have two equal real number roots? Find these two equal real roots.

The answer ⊿=b -4ac= 16-4(3-k)=4+4k. Because the equation has two equal real roots, ⊿=0, so 4+4k=0 k=- 1. If we substitute it into the original equation, we get: x-4x+4 = 0.

10. (XX Xinjiang Construction Corps, Xinjiang Uygur Autonomous Region) Solve the equation: 2x2-7x+6=0.

Answer solution:

1 1.(XX Foshan, Guangdong) There will be such a problem in textbooks or materials: turn the equation into a general form of a quadratic equation with one variable, and write its quadratic coefficient, linear coefficient and constant term.

Now adapt the above questions into the following two small questions. Please answer them.

(1) Which of the following equations is the general form of a quadratic equation with one variable? (The answer is only serial number).

① ② ③

④ ⑤

(2) What is the relationship among quadratic coefficient, linear coefficient and constant term after the equation is transformed into a general quadratic equation?

Answer: (1) answer: 1245 (each 1) .......................................................................................................................................

(2) If its quadratic coefficient is a(a≠0), the coefficient of the first term is -2a, and the constant term is -2a. ............................................................................................................................