Equation and inequations of the second degree: exercises in 1ère.

The chapter on equation and inequation is very important in Math and it helps the student to progress well. With a series of math exercises in 1st S on equations and inequations of the second degree, the student will have the opportunity to practice more.

You will find in these corrected exercises of maths in first degree on equations and inequations of the second degree, the following concepts:

canonical form;
method of resolution with the delta discriminant;
solve a second degree inequation with one unknown;
solve an inequation by the graphical method.

Exercise 1: equation

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

2) Let f be the function defined on $\mathbb{R}$ by .

Determine the number of real solutions of the equation f(x)=0.

Exercise 2:

determine the possible root(s) of the following functions.

$b)g(x)=x^2+x+\frac{1}{4}.$

2) Determine all real solutions of the following equations.

Exercise 3:

Let f be the function defined on $\mathbb{R}$ by $f(x)=-7x^2-\frac{49}{2}x+14$ .

determine the number of roots of the function f, justifying.
Verify that – 4 is a root of f.
Using the sum or product of the roots, determine the value of the other root.

Exercise 4:

For each function below, determine if it is a polynomial function of degree 2.

$a)f(x)=x^2+2x-\sqrt{2}$ .

$b)g(x)=x^2+\frac{1}{x}-1$ .

Exercise 5:

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

1)Expand the expression .

2)Deduce the canonical form of f.

Exercise 6:

Determine the canonical form of the following functions.

Exercise 7:

For each function shown below,

determine the coordinates of the vertex, the axis of symmetry and the sign of .

Exercise 8: equation

For each parabola equation given below,

determine its axis of symmetry and the coordinates of the vertex.

Exercise 9: equation

Factor the following expressions, using remarkable identities.

Exercise 10: equation

For each trinomial below, calculate the discriminant $\Delta$ .

Exercise 11: equation

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

$c)2x^2-2x+\frac{1}{2}=0.$

Exercise 12:

Solve in $\mathbb{R}$ the following inequalities without using the discriminant.

$b)x^2-81\leq\,\,0.$

Exercise 13:

Let f be a polynomial function of degree 2 defined on $\mathbb{R}$ by
$f(x)\,=\,ax^2\,+\,bx\,+\,c$ with $a\neq\,0$ .

The representative curve of f is given below.

1. Using the coordinates of point A, determine the value of c.
2. Using the coordinates of points B and C, determine the value
of the coefficients a and b.

3. Derive the expression of f(x) as a function of x.

Exercise 14:

For each trinomial below, calculate the discriminant $\Delta$ .

Exercise 15: equation

Determine the number of real solutions of each equation below.
$a)\,x^2+\,3x+2=0$

$c)-\frac{1}{3}x^2+2x-3=0$

Exercise 16:

For each trinomial shown graphically below, determine the sign of $\Delta$ .

Exercise 17:

Draw the sign table for each function defined below.
$a)\,f(x)\,=\,2x^2-\,4x-\,16$
$b)\,g(x)\,=\,9x^2\,+\,24x+\,16$
$c)\,h(x)\,=\,2x^2-\,5x+6$

Exercise 18:

Solve in $\mathbb{R}$ the following inequalities without using the discriminant.

$b)x^2-81\leq\,\,0$

Exercise 19:

Solve in $\mathbb{R}$ the following inequalities.
$a)\,3x^2\,-4x+\frac{4}{3}\leq\,\,0$

Determine the set of real solutions of the following inequalities.
$a)-6x^2+\,15x-4\,\leq\,\,2$
$b)\,-7x^2+4x-9>-8$

Exercise 20:

Let fla be defined on $\mathbb{R}$ by $f(x)\,=x^2-6x-\,27$ .
1. Determine the canonical form of f, using the remarkable identities.
2. Determine the factorized form of f, using the remarkable identities.
3. Using the appropriate form, solve for:

$a)\,f(x)=0\\\,b)\,f(x)=-27\\\,c)\,f(x)=-36$

4.let g be the function defined on $\mathbb{R}$ by $g(x)\,=\,2x^2-\frac{3}{2}x-\frac{1}{2}$ .
a) Verify that 1 is the root of g.
b) Using the sum or product of the roots determine the value of the other root of g.
5. solve for f(x) < g(x).

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Equations and inequations of the second degree: math exercises in 1st grade corrected in PDF.