5x^2+3x-9=0
1) Let's solve in ℝ: 5x2 + 3x -9 = 0
Given the quadratic form ax2 + bx + c, you can calculate the discriminant: Δ = b2 -4ac. Here a = 5, b = 3 and c = -9
Δ = 32 -4·5·(-9)
ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product: 32 = 3·3 = 9■Multiplications are done before additions and subtractions: 4·5 = 20■Two minus signs cancel out, the product of two negative numbers is positive: -20·(-9) = 20·9
Δ = 9 + 20·9
ⓘ Multiplications are done before additions and subtractions: 20·9 = 180
Δ = 9 + 180
ⓘ Start from 9 and move forward 180 on the number line: 9 + 180 = 189
Δ = 189
As Δ > 0, the equation has two roots:
The quadratic formula gives x1 = -b -Δ2a = -3 -1892·5
x1 = -3 -1892·5
ⓘ We look for the largest square dividing 189, 9 = 32, and take it out of the root: 189 = 9·21 = 9·21 = 3·21■The minus sign is factored out, changing the sign of each term: -3 -3·21 = -(3 + 3·21)
ⅰ 2·5 = 10
x1 = -3 + 3·2110
The quadratic formula gives x2 = -b + Δ2a = -3 + 1892·5
x2 = -3 + 1892·5
ⓘ We look for the largest square dividing 189, 9 = 32, and take it out of the root: 189 = 9·21 = 9·21 = 3·21
ⅰ 2·5 = 10
x2 = -3 + 3·2110
▶Solutions obtained: x = { -3 + 3·2110, 3·21 -310 }
2) The numeric value of: x = { -3 + 3·2110, 3·21 -310 } equals:
x = { ≈ -3 + 3·4.582610, ≈ 3·4.5826 -310 }
x = { ≈ -3 + 13.747710, ≈ 13.7477 -310 }
x = { ≈ -16.747710, ≈ 10.747710 }
▷ x = { ≈ -1.6748, ≈ 1.0748 }