factor -x^2-5x+24 — step-by-step solution

Let's factor in

ℝ

- x^{2} - 5 x + 24

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 = -5

and

c = 24

Δ = {(-5)}^{2} - 4 \cdot (-1) \cdot 24

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

{(-5)}^{2}

(-5) \cdot (-5)

25

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

-4 \cdot (-1)

4

Δ = 25 + 4 \cdot 24

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 24

4 \cdot (20 + 4)

4 \cdot 20 + 4 \cdot 4

80 + 16

96

Δ = 25 + 96

ⓘ Start from

25

and move forward

96

on the number line:

25 + 96

121

Δ = 121

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-5) - \sqrt{121}}{2 \cdot (-1)}

x_{1} = \frac{5 - \sqrt{121}}{2 \cdot (-1)}

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

\sqrt{121}

11

;

11^{2}

121

■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{5 - 11}{-2}

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

\frac{5 - 11}{-2}

- \frac{5 - 11}{2}

x_{1} = - \frac{5 - 11}{2}

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

5 - 11

-6

x_{1} = - (\frac{-6}{2})

ⓘ Two minus signs cancel out:

- (\frac{-6}{2})

\frac{6}{2}

x_{1} = \frac{6}{2}

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

2

, their greatest common divisor:

\frac{6}{2}

3

x_{1} = 3