Step-by-step explanation:
the area of a triangle is
baseline × height / 2
so, in our case we know this is 10 :
(x+3)(x-1)/2 = 10
(x+3)(x-1) = 20
x² - x + 3x - 3 = 20
x² + 2x - 23 = 0
that is exactly the provided equation. so, yes, x satisfies it.
the general solution to a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = 2
c = -23
x = (-2 ± sqrt(2² - 4×1×-23))/(2×1) =
= (-2 ± sqrt(4 + 92))/2 = (-2 ± sqrt(96))/2 =
= (-2 ± sqrt(16×6))/2 = (-2 ± 4×sqrt(6))/2 =
= -1 ± 2×sqrt(6)
x1 = -1 + 2×sqrt(6) = 3.898979486... ≈ 4
or using sqrt(6) ≈ 2.45
= -1 + 2×2.45 = -1 + 4.9 = 3.9 ≈ 4
x2 = -1 - 2×sqrt(6) = -5.898979486... ≈ -6
or using sqrt(6) ≈ 2.45
= -1 - 2×2.45 = -1 - 4.9 = -5.9 ≈ -6
x2 would give us negative lengths for the triangle, which does not make sense.
so, x = 4 is our solution.
and that makes the height x-1 = 4-1 = 3 units.
