2x^2-16x=-30 — step-by-step solution

Let's solve in

ℝ

2 x^{2} - 16 x = -30

Since

2 \neq 0

, divide both sides by

2

x^{2} - 8 x = -15

x^{2} - 8 x + 15 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -8

and

c = 15

Δ = {(-8)}^{2} - 4 \cdot 1 \cdot 15

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

{(-8)}^{2}

(-8) \cdot (-8)

64

Δ = 64 - 4 \cdot 15

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 15

4 \cdot (10 + 5)

4 \cdot 10 + 4 \cdot 5

40 + 20

60

Δ = 64 - 60

ⓘ Start from

64

and move back

60

on the number line:

64 - 60

4

Δ = 4

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-8) - \sqrt{4}}{2 \cdot 1}

x_{1} = \frac{8 - \sqrt{4}}{2}

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

\sqrt{4}

2

;

2^{2}

4

x_{1} = \frac{8 - 2}{2}

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

8 - 2

6

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

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-8) + \sqrt{4}}{2 \cdot 1}

x_{2} = \frac{8 + \sqrt{4}}{2}

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

\sqrt{4}

2

;

2^{2}

4

x_{2} = \frac{8 + 2}{2}

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

8 + 2

10

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

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

2

, their greatest common divisor:

\frac{10}{2}

5

x_{2} = 5

▶Solutions found:

x = {3, 5}