8r^2+3r+2=7r^2 — step-by-step solution

Let's solve in

ℝ

8 r^{2} + 3 r + 2 = 7 r^{2}

8 r^{2}

+ 3 r

+ 2

- 7 r^{2} = 0

ⓘ Like terms are grouped by adding their coefficients:

8 r^{2} - 7 r^{2}

r^{2}

r^{2} + 3 r + 2 = 0

Given the quadratic form

a r^{2} + b r + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 3

and

c = 2

Δ = 3^{2} - 4 \cdot 1 \cdot 2

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

3^{2}

3 \cdot 3

9

Δ = 9 - 4 \cdot 2

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 2

8

Δ = 9 - 8

ⓘ Start from

9

and move back

8

on the number line:

9 - 8

1

Δ = 1

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

r_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 3 - \sqrt{1}}{2 \cdot 1}

r_{1} = \frac{- 3 - 1}{2}

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

- 3 - 1

-4

r_{1} = - \frac{4}{2}

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

2

, their greatest common divisor:

- \frac{4}{2}

-2

r_{1} = -2

The quadratic formula gives

r_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 3 + \sqrt{1}}{2 \cdot 1}

r_{2} = \frac{- 3 + 1}{2}

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

- 3 + 1

-2

r_{2} = - \frac{2}{2}

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

2

, their greatest common divisor:

- \frac{2}{2}

-1

r_{2} = -1

▶Solutions found:

r = {-2, -1}