x^2+11x+28<0 — step-by-step solution

Let's solve in

ℝ

x^{2} + 11 x + 28 < 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

Δ = 11^{2} - 4 \cdot 1 \cdot 28

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

11^{2}

11 \cdot 11

121

Δ = 121 - 4 \cdot 28

ⒾMultiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

- 4 \cdot 28

- 4 \cdot (20 + 8)

- (4 \cdot 20 + 4 \cdot 8)

- (80 + 32)

-112

Δ = 121 - 112

ⒾStart from

121

and move back

112

on the number line:

121 - 112

9

Δ = 9

Since

Δ > 0

, the equation has two roots:

For the first root, we obtain:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{- 11 - \sqrt{9}}{2}

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

- \sqrt{9}

-3

;

3^{2}

9

x1 = \frac{- 11 - 3}{2}

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

- 11 - 3

-14

x1 = - \frac{14}{2}

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

2

, their greatest common divisor:

- \frac{14}{2}

-7

x1 = -7

The second root is:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{- 11 + \sqrt{9}}{2}

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

\sqrt{9}

3

;

3^{2}

9

x2 = \frac{- 11 + 3}{2}

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

- 11 + 3

-8

x2 = - \frac{8}{2}

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

2

, their greatest common divisor:

- \frac{8}{2}

-4

x2 = -4

The quadratic has the opposite sign to a =

1

between its roots, so it is indeed negative between its roots; the roots, where it vanishes, are excluded.

▶Solution obtained:

x \in (-7, -4)