Equations and inequations of the second degree : 11th worksheets

11th grade math worksheets

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Second degree equations and inequations with online 11th grade math worksheets to progress in high school math.

Exercise #1:

Determine the canonical form of the following polynomials:

a. $\,P_1(x)=3x^2-4x+6$

b. $\,P_2(x)=0,5x^2-4x+3$

c. $\,P_3(x)=5x^4+6x^2+1$

d. $\,P_4(x)=4x^2-12x+9$

e. $\,P_5(x)=x^2+ax+a\,\,\,(a\in\mathbb{R})$

Exercise 2:

Solve the following equations in $\,\mathbb{R}$ .

a. $\,4x^2-7=0$

b. $\,-9x^2+3x=0$

c. $\,7(x-1)^2-6=0$

d. $\,4(x-1)(2x+3)-(4x-1)(5x+2)=0$

e. $\,4(x-2)^2=9(2x+3)^2$

f. $\,2x^2+7x-9=0$

g. $\,\frac{1}{2}x^2+5x+\frac{25}{2}=0$

h. $\,(x^2-2x-3)(x^2+2x+2)=0$

i. $\,x^4-2x^2+1=0$

Exercise #3:

Determine a and b two real numbers such that :

1. $\,2x^3-2x^2-x+1=(x-1)(ax^2-b)$

2. $\,\frac{4x+23}{x+7}=a+\frac{b}{x+7}$

Exercise #4:

Solve in $\,\mathbb{R}$ the following inequalities:

a. $\,4-3x\ge\,0$

b. $\,(2x+3)^2>9$

c. $\,9x^2-6x+1<0$

d. $\,x^2\,\ge\,-2x-7$

e. $\,x^2\,\ge\,-2x-7$

Exercise #5:

Determine the domain of definition of the following functions defined by :

a. $\,f(x)=\frac{2x-1}{x^3+x^2-2x}$ .

b. $\,g(x)=3-\sqrt{2x^4+5x^3-3x^2}$ .

c. $\,h(x)=\sqrt{\frac{x^2-2x-15}{x^2-x-2}}$ .

Exercise 6 – Equations in fraction form
Solve the following equations and inequations:

1. $\frac{5x+4}{x-1}=0$ .

2. $\frac{2x^2-5x}{x+4}=x$ .

3. $(4x-5)^2\geq\,\,(6x+1)^2$ .

Exercise 7 – Solving a complex inequation

Solve the following inequation:

$\frac{(6x+8-2x^2)(3-x)}{4x^2-12x+9}\geq\,\,0$

Exercise 8 – Trinomial and second degree equations

The questions in this exercise are independent.

1. Find a polynomial function of the second degree with 1 and 4 as roots.

Can we find such a polynomial whose representative curve passes through the point A(-1; 1 )?

If yes, give it.

2. Show that for a real , the equation has two distinct real solutions.

3. is a real number.Consider equation (E): .

For what values of does equation (E) have a unique solution?

Calculate this solution.

Exercise 9 – Geometry

In a 10 cm square, we colored a strip of width x cm and a square of side x cm centered as on the figure below.

1. Express the area of the colored part as a function of x. Let’s note A(x) this area.

2. Determine for which value of x, the colored area is equal to the area of the white part.

Exercise 10 – Equations and inequations to solve

Solve the following equations and inequations:

3. $\frac{x}{x-1}-\frac{3x-4}{(x-1)(x-2)}=0$

4. $\frac{(2x^2+x-3)(x^2+5)}{8-4x}\geq\,,0$

Exercise 11 – Find the perimeter of a right triangle
The area of a right triangle is 429 m², and the hypotenuse has length h = 72.5 m.
Find the perimeter.
Exercise 12 – Cyclist Speeds
To get from city A to city B, which is 195 km away, two cyclists leave at the same time.
One of them, whose average speed on this route is 4 km/h higher than the other, arrives one hour earlier.

What are the speeds of the two cyclists?

Exercise 13 – Problem and equations of the second degree
1. We have a 10 cm long wooden stick.
Where to break the stick so that the pieces obtained are two consecutive sides of a rectangle of surface 20 cm²?

2. Same question with a rectangle of area 40 cm².

Exercise 14 – Solving second degree inequalities

Solve the following inequations:

1. $-2x^2+7x-5\leq\,\,0$

3. $\frac{3x^2+x+1}{x^2-3x-10}>0$

Exercise 15 – Change of variable

1. Solve the following equations:

a. $x^2=\frac{1}{2}$ .

b. $x^2=\frac{1}{3}$ .

2. Solve the following equation: .

Hint: we can ask
Exercise 16 – Rectangular flag

On a rectangular flag of length 4 m and width 3 m, there is a cross of thickness x m.
What value must be given to the width of the cross so that its area is equal to half the flag?

Exercise 17 – Solving second degree equations
Solve in $\mathbb{R}$ the following equations:

1. $x^2\,=\,9.$

2. $x^2\,=\,-3$ .

3. $(x\,-\,5)^2\,=\,3$ .

4. $(2x\,-1)^2\,+\,x(1\,-\,2x)\,=\,4x^2\,-1$ .

5) $(3x\,+\,5)^2\,=\,(x\,+\,1)^2$ .

6) $(5x\,-\,4)^2\,-\,(3x\,+\,7)^2\,=\,0.$

Exercise 18 – Finding three squares

Can we find three squares with consecutive integer sides whose sum of areas is equal to 15,125?
If so, specify what values the sides should have.
Same question with 15 127.

Exercise 19 – Geometry

In a triangle ABC rectangular in A, we place the points D and E respectively on [AC] and [AB] such that
AD = BE = x.

Determine x so that the area of triangle ADE is half that of triangle ABC.

We give: AB = 18 m and AC = 8 m.

Exercise 20 – Solving equations and inequalities

Solve:

b. $2(7x-1)\,<\,4x-3$

c. $(x-1)(2-x)(x-3)(4-x)\leq\,\,0$

Exercise 21 – Equations and fractions

Solve the following equations:

1. $\frac{2x-5}{x-1}=\frac{x-1}{x+1}$

2. $\frac{x^2-x+1}{x+2}=2x+3$

Exercise 22 – Gasoline Prices

One buys for 40 € of gasoline at a service station.
We realize that at another station, the price of a liter of gasoline is 0.10 € lower.
We could have obtained 5 more liters for the same price.

What was the price of gas at the first station and how many liters did we take?

Exercise 23 – Finding two numbers
Find two numbers whose sum is equal to 57 and whose product is equal to 540.

Exercise 24 – Solving an inequation graphically and by calculation

Solve the following inequation:
$-x^2\,+\,9x\,+\,22\,>\,0.$

Solve the following equations:
1) $4x^2\,-\,x\,-\,3\,=\,0.$

2) $(t\,+\,1)^2\,+\,3\,=\,0.$

3) $2(2x\,+\,1)^2\,-(2x\,+\,1)-\,6\,=\,0.$

4) $x^2\,+\,10^{50}\,+\,25\times \,10^{98\,}=\,0$ .
Exercise 25 – Roots and functions of the second degree

Consider the function defined on $\mathbb{R}$ by .

1. Determine the possible roots of and derive, if possible, a factorization of .

2. Determine the coordinates of the intersection points of the curve of with the axes of the reference frame.

3. Let be the function defined on $\mathbb{R}$ by .

Study the relative position of the curves of and .

Exercise 26 – Study of a trinomial

We give the second degree trinomial P defined by :

$P(x)=4x^2-(\sqrt{6}+4\sqrt{3})x+\sqrt{18}$

1. Show that P has $\frac{\sqrt{6}}{4}$ as its root.

2. Find the other root (in exact value).

Exercise 27 – Equations of the second degree

Solve the following equations:

$b)\,x^2\,=\,-\,3$

$c)\,(x-5)^2\,=3$

Exercise 28 – Geometry

In a triangle ABC rectangular in A, we place the points D and E respectively on [AC] and [AB] such that AD=BE=x.

Determine x so that the area of triangle ADE is half that of triangle ABC.

Data: AB= 18 m and AC = 8 m.

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