-14x-45>x^2 — step-by-step solution

Let's solve in

ℝ

- 14 x - 45 > x^{2}

- x^{2} - 14 x - 45 > 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = -1

b = -14

and

c = -45

Δ = {(-14)}^{2} - 4 \cdot (-1) \cdot (-45)

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

{(-14)}^{2}

(-14) \cdot (-14)

196

■Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-4 \cdot (-1)

4

■The minus sign is placed in front of the product:

4 \cdot (-45)

- 4 \cdot 45

Δ = 196 - 4 \cdot 45

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 45

4 \cdot (40 + 5)

4 \cdot 40 + 4 \cdot 5

160 + 20

180

Δ = 196 - 180

ⓘ Start from

196

and move back

180

on the number line:

196 - 180

16

Δ = 16

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-14) - \sqrt{16}}{2 \cdot (-1)}

x_{1} = \frac{14 - \sqrt{16}}{2 \cdot (-1)}

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

\sqrt{16}

4

;

4^{2}

16

■A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-1)

-2

x_{1} = \frac{14 - 4}{-2}

ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{14 - 4}{-2}

- \frac{14 - 4}{2}

x_{1} = - \frac{14 - 4}{2}

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

14 - 4

10

x_{1} = - \frac{10}{2}

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

2

, their greatest common divisor:

- \frac{10}{2}

-5

x_{1} = -5