Solve the Quadratic Inequalities

The best Maths tutors available
1st lesson free!
Intasar
4.9
4.9 (26 reviews)
Intasar
£36
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Paolo
4.9
4.9 (13 reviews)
Paolo
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (7 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Ayush
5
5 (28 reviews)
Ayush
£60
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Farooq
5
5 (14 reviews)
Farooq
£40
/h
1st lesson free!
Tom
5
5 (9 reviews)
Tom
£22
/h
1st lesson free!
Intasar
4.9
4.9 (26 reviews)
Intasar
£36
/h
1st lesson free!
Matthew
5
5 (17 reviews)
Matthew
£25
/h
1st lesson free!
Paolo
4.9
4.9 (13 reviews)
Paolo
£25
/h
1st lesson free!
Dr. Kritaphat
4.9
4.9 (7 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!
Ayush
5
5 (28 reviews)
Ayush
£60
/h
1st lesson free!
Petar
4.9
4.9 (9 reviews)
Petar
£27
/h
1st lesson free!
Farooq
5
5 (14 reviews)
Farooq
£40
/h
1st lesson free!
Tom
5
5 (9 reviews)
Tom
£22
/h
First Lesson Free>

Exercise 1

x^2 - 6x + 8 > 0

Exercise 2

x^2 + 2x +1 \geq 0

Exercise 3

x^2 + x +1 > 0

Exercise 4

7x^2 + 21x - 28 < 0

Exercise 5

−x^2 + 4x - 7 < 0

Exercise 6

4x^2-16- 0 \geq 0

Exercise 7

4x^2-16-0 \leq 0

Exercise 8

x^4+12x^3-64x^2>0

Exercise 9

x^4 - 25x^2+ 144 < 0

Exercise 10

x^4- 16x^2 - 225 \geq 0

 

Solution of exercise 1

x^2 - 6x + 8 > 0

x^2 - 6x + 8 = 0

x = \frac{6 \pm \sqrt {-6^2  - 4.8}} {2}

x = \frac {6 \pm \sqrt { 36 - 32}} {2}

=\frac{6 \pm 2} {2}

x_1 =\frac {8}{2}=4

x_2 =\frac {4}{2}=2

P(0) = 0^2 - 6 · 0 + 8 > 0

P(3) = 3^2 - 6 · 3 + 8 = 17 - 18 < 0

P(5) = 5^2 - 6 · 5 + 8 = 33 - 30 > 0

Solution set of exercise 1 on a number line

S = (-\infty, 2) (4, \infty)

 

Solution of exercise 2

x^2 + 2x +1 = 0

x = \frac {-2 \pm \sqrt{2^2 - 4}}{2}

x = \frac {-2\pm 0}{2}

x=1

(x + 1)^2 \geq 0

As a number squared is always positive.

S =

 

Solution of exercise 3

x^2+ x +1 > 0

x^2+ x +1 = 0

x = \frac {-1 \pm \sqrt {1-4}}{2}

x = \frac {-1 \pm \sqrt{-3}}{2}

P(0) = 0 + 0 + 1 > 0

The sign obtained coincides with the inequality, the solution is .

 

Solution of exercise 4

7x^2 + 21x - 28 < 0

x^2 +3x - 4 < 0

x^2 +3x - 4 = 0

x = \frac {-3 \pm \sqrt {9+16}}{2}

x = \frac{-3 \pm 5}{2}

x_1 = 1

x_2 = -4

P(-6) = (-6)^2 + 3 (-6) - 4 > 0

P(0) = 0^2 + 3(0) - 4 < 0

P(3) = 3^2 + 2 (3) - 4 > 0

Solution set of exercise 4 on a number line

(-4,1)

 

Solution of exercise 5

−x^2 + 4x - 7 < 0

x^2 - 4x + 7 = 0

x = \frac {4 \pm \sqrt {16-28}}{2}

x = \frac {4 \pm \sqrt {-12}}{2} \notin R

P(0) = -0^2 +4(0) - 7 < 0

S = R

 

Solution of exercise 6

4x^2 -16 \geq 0

4x ^2 = 16

x^2 = 4x = \pm \sqrt {4}

x_1 = 2

x_2 = -2

Solution of exercise 6

P(-3) = 4 · (-3)^2 - 16 > 0

P(0) = 4 · 0 ^2 - 16 < 0

P(3) = 4 · 3 ^2 - 16 > 0

Solution of exercise 6

(-\infty , -2] U [2 + \infty)

Solution of exercise 7

4x^2 - 4x + 1 \leq 0

4x^2 - 4x + 1 = 0

x = \frac{4 \pm \sqrt {16-16}}{8}

x = \frac{4}{8}

x = \frac {1}{2}

Solution of exercise 8

x^4 +12x^3 -64x^2>0

x^2(x^2 +12x -64)>0

The first factor is always positive.

x^2 +12x -64 =0

x = \frac {-12 \pm \sqrt {144+256}}{2}

x = \frac {-12 \pm 20}{2}

x_2 = 4

x_3 = -16

Solution set of exercise 8

P(−17) = (−17) ^2+ 12 · 17 - 64 > 0

P(0) = 0^2 + 12 · 0 - 64 < 0

P(5) = 5 ^2 + 12 · 5 - 64 > 0

Solution set of exercise 8 on a number line

(\infty, -16] U [4, \infty)

 

Solution of exercise 9

x^4 - 25x^2 + 144 < 0

x^ 4 - 25x^2+ 144 = 0

x^2=t

t = \frac {25 \pm \sqrt {625-576}}{2}

t = \frac {25 \pm 7}{2}

t_1 = 16

t_2 = 9

x ^ 2 = 16

x = \pm \sqrt {16}

x_1 = 4

x_2 = -4

x^2=9

x = \pm \sqrt {16}

x_1 = 4

x_2 = -4

Solution set of exercise 9 on a number line

(-4, -3) U (3, 4)

 

Solution of exercise 10

x^4 - 16x^2 - 225 \geq 0

x^4 - 16x^2 - 225 = 0

x^2 = t

t^2 - 16t - 225 = 0

t = \frac {16 \pm \sqrt {256+900}}{2}

t = \frac 16 \pm 34}{2}

t_1 = 25

t_2 = -9

x^2 = 25

x = \pm \sqrt{25}

x_1 = 5

x_2 = -5

x^2 = -9

x = \pm \sqrt{-9} \notin R

(x^2 - 25) · (x^2 + 9) \geq 0

The second factor is always positive and different to 0.

(x^2 - 25) \geq 0

Solution set of exercise 10 on a number line

(-\infty, -5] U [5, +\infty)

 

Need a Maths teacher?

Did you like the article?

1 Star2 Stars3 Stars4 Stars5 Stars 5.00/5 - 2 vote(s)
Loading...

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.