factor -91x^2+108x-32 — step-by-step solution

Let's factor in

ℝ

- 91 x^{2} + 108 x - 32

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = -91

b = 108

and

c = -32

Δ = 108^{2} - 4 \cdot (-91) \cdot (-32)

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

108^{2}

108 \cdot 108

11,664

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

-4 \cdot (-91)

4 \cdot 91

4 \cdot (90 + 1)

4 \cdot 90 + 4

360 + 4

364

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

364 \cdot (-32)

- 364 \cdot 32

Δ = 11,664 - 364 \cdot 32

ⓘ Multiplications are done before additions and subtractions:

364 \cdot 32

364 \cdot (30 + 2)

364 \cdot 30 + 364 \cdot 2

10,920 + 728

11,648

Δ = 11,664 - 11,648

ⓘ Start from

11,664

and move back

11,648

on the number line:

11,664 - 11,648

16

Δ = 16

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 108 - \sqrt{16}}{2 \cdot (-91)}

x_{1} = \frac{- 108 - \sqrt{16}}{2 \cdot (-91)}

ⓘ 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 (-91)

- 2 \cdot 91

- 2 \cdot (90 + 1)

- (2 \cdot 90 + 2)

- (180 + 2)

-182

x_{1} = \frac{- 108 - 4}{-182}

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

\frac{- 108 - 4}{-182}

- (\frac{- 108 - 4}{182})

x_{1} = - (\frac{- 108 - 4}{182})

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

- 108 - 4

-112

x_{1} = - (\frac{-112}{182})

ⓘ Two minus signs cancel out:

- (\frac{-112}{182})

\frac{112}{182}

x_{1} = \frac{112}{182}

We divide the numerator and denominator of

\frac{112}{182}

by the common factor:

14

, which gives the new quotient:

\frac{8}{13}

x_{1} = \frac{8}{13}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 108 + \sqrt{16}}{2 \cdot (-91)}

x_{2} = \frac{- 108 + \sqrt{16}}{2 \cdot (-91)}

ⓘ 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 (-91)

- 2 \cdot 91

- 2 \cdot (90 + 1)

- (2 \cdot 90 + 2)

- (180 + 2)

-182

x_{2} = \frac{- 108 + 4}{-182}

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

\frac{- 108 + 4}{-182}

- (\frac{- 108 + 4}{182})

x_{2} = - (\frac{- 108 + 4}{182})

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

- 108 + 4

-104

x_{2} = - (\frac{-104}{182})

ⓘ Two minus signs cancel out:

- (\frac{-104}{182})

\frac{104}{182}

x_{2} = \frac{104}{182}

We simplify the fraction:

\frac{104}{182}

by the common factor:

26

to get the new quotient:

\frac{4}{7}

x_{2} = \frac{4}{7}

A trinomial with roots

x_{1}

and

x_{2}

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

; bringing the coefficient into the brackets:

-91 \cdot (x - \frac{4}{7}) \cdot (x - \frac{8}{13})

- (7 x - 4) \cdot (13 x - 8)

▶Factors identified:

- (7 x - 4) \cdot (13 x - 8)