h^2+30h+200=0 — step-by-step solution

Let's solve in

ℝ

h^{2} + 30 h + 200 = 0

Given the quadratic form

a h^{2} + b h + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 30

and

c = 200

Δ = 30^{2} - 4 \cdot 1 \cdot 200

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

30^{2}

30 \cdot 30

900

Δ = 900 - 4 \cdot 200

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 200

800

Δ = 900 - 800

ⓘ Start from

900

and move back

800

on the number line:

900 - 800

100

Δ = 100

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

h_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 30 - \sqrt{100}}{2 \cdot 1}

h_{1} = \frac{- 30 - \sqrt{100}}{2}

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

\sqrt{100}

10

;

10^{2}

100

h_{1} = \frac{- 30 - 10}{2}

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

- 30 - 10

-40

h_{1} = - \frac{40}{2}

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

2

, their greatest common divisor:

- \frac{40}{2}

-20

h_{1} = -20

The quadratic formula gives

h_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 30 + \sqrt{100}}{2 \cdot 1}

h_{2} = \frac{- 30 + \sqrt{100}}{2}

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

\sqrt{100}

10

;

10^{2}

100

h_{2} = \frac{- 30 + 10}{2}

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

- 30 + 10

-20

h_{2} = - \frac{20}{2}

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

2

, their greatest common divisor:

- \frac{20}{2}

-10

h_{2} = -10

▶Solutions found:

h = {-20, -10}