2x^2+11x+9=10x+10 — step-by-step solution

1) Let's solve in

ℝ

2 x^{2} + 11 x + 9 = 10 x + 10

2 x^{2}

+ 11 x

+ 9

- 10 x

- 10 = 0

ⓘ Start from

9

and move back

10

on the number line, going below zero:

9 - 10

-1

■Like terms are grouped by adding their coefficients:

11 x - 10 x

x

2 x^{2} + x - 1 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 2

b = 1

and

c = -1

Δ = 1 - 4 \cdot 2 \cdot (-1)

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 2

8

■Multiplying by -1 gives the opposite:

- 8 \cdot (-1)

8

Δ = 1 + 8

ⓘ Start from

1

and move forward

8

on the number line:

1 + 8

9

Δ = 9

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 1 - \sqrt{9}}{2 \cdot 2}

x_{1} = \frac{- 1 - \sqrt{9}}{2 \cdot 2}

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

\sqrt{9}

3

;

3^{2}

9

ⅰ

2 \cdot 2

4

x_{1} = \frac{- 1 - 3}{4}

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

- 1 - 3

-4

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

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

4

, their greatest common divisor:

- \frac{4}{4}

-1

x_{1} = -1

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 1 + \sqrt{9}}{2 \cdot 2}

x_{2} = \frac{- 1 + \sqrt{9}}{2 \cdot 2}

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

\sqrt{9}

3

;

3^{2}

9

ⅰ

2 \cdot 2

4

x_{2} = \frac{- 1 + 3}{4}

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

- 1 + 3

2

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

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

2

, their greatest common divisor:

\frac{2}{4}

\frac{1 \cdot 2}{2 \cdot 2}

\frac{1}{2}

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

▶Solutions found:

x = {-1, \frac{1}{2}}

2) The numeric value of:

x = {-1, \frac{1}{2}}

equals:

▷

x = {-1, 0.5}