2x^2+17x+21<= 0 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} + 17 x + 21 \leq 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = 17

and

c = 21

Δ = 17^{2} - 4 \cdot 2 \cdot 21

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

17^{2}

17 \cdot 17

289

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 289 - 8 \cdot 21

ⓘ Multiplications are done before additions and subtractions:

8 \cdot 21

8 \cdot (20 + 1)

8 \cdot 20 + 8

160 + 8

168

Δ = 289 - 168

ⓘ Start from

289

and move back

168

on the number line:

289 - 168

121

Δ = 121

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 17 - \sqrt{121}}{2 \cdot 2}

x_{1} = \frac{- 17 - \sqrt{121}}{2 \cdot 2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{121}

11

;

11^{2}

121

ⅰ

2 \cdot 2

4

x_{1} = \frac{- 17 - 11}{4}

ⓘ The numerator and the denominator are computed before the division:

- 17 - 11

-28

x_{1} = - \frac{28}{4}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

4

, their greatest common divisor:

- \frac{28}{4}

-7

x_{1} = -7

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 17 + \sqrt{121}}{2 \cdot 2}

x_{2} = \frac{- 17 + \sqrt{121}}{2 \cdot 2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{121}

11

;

11^{2}

121

ⅰ

2 \cdot 2

4

x_{2} = \frac{- 17 + 11}{4}

ⓘ The numerator and the denominator are computed before the division:

- 17 + 11

-6

x_{2} = - \frac{6}{4}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{6}{4}

- \frac{3 \cdot 2}{2 \cdot 2}

- \frac{3}{2}

x_{2} = - \frac{3}{2}

The quadratic has the opposite sign to a =

2

between its roots, so it is indeed negative between its roots; the roots, where it is zero, are included.

▶Solution obtained:

x \in [-7, - \frac{3}{2}]

2) The numeric value of:

x \in [-7, - \frac{3}{2}]

is:

▷

x \in [-7, -1.5]