factor 5x^2-43x+24 — step-by-step solution

Let's factor in

ℝ

5 x^{2} - 43 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 = 5

b = -43

and

c = 24

Δ = {(-43)}^{2} - 4 \cdot 5 \cdot 24

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

{(-43)}^{2}

(-43) \cdot (-43)

1849

■Multiplications are done before additions and subtractions:

4 \cdot 5

20

Δ = 1849 - 20 \cdot 24

ⓘ Multiplications are done before additions and subtractions:

20 \cdot 24

20 \cdot (20 + 4)

20 \cdot 20 + 20 \cdot 4

400 + 80

480

Δ = 1849 - 480

ⓘ Start from

1849

and move back

480

on the number line:

1849 - 480

1369

Δ = 1369

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-43) - \sqrt{1369}}{2 \cdot 5}

x_{1} = \frac{43 - \sqrt{1369}}{2 \cdot 5}

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

\sqrt{1369}

37

;

37^{2}

1369

ⅰ

2 \cdot 5

10

x_{1} = \frac{43 - 37}{10}

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

43 - 37

6

x_{1} = \frac{6}{10}

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

2

, their greatest common divisor:

\frac{6}{10}

\frac{3 \cdot 2}{5 \cdot 2}

\frac{3}{5}

x_{1} = \frac{3}{5}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-43) + \sqrt{1369}}{2 \cdot 5}

x_{2} = \frac{43 + \sqrt{1369}}{2 \cdot 5}

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

\sqrt{1369}

37

;

37^{2}

1369

ⅰ

2 \cdot 5

10

x_{2} = \frac{43 + 37}{10}

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

43 + 37

80

x_{2} = \frac{80}{10}

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

10

, their greatest common divisor:

\frac{80}{10}

8

x_{2} = 8

A trinomial with roots

x_{1}

and

x_{2}

a \cdot (x - x_{1}) \cdot (x - x_{2})

; bringing the coefficient into the brackets:

5 \cdot (x - 8) \cdot (x - \frac{3}{5})

(x - 8) \cdot (5 x - 3)

▶Factors identified:

(x - 8) \cdot (5 x - 3)