x^3-7x^2+10x<0 — step-by-step solution

⚠Ⓘ 7x is read as the product 7*x: for a power, write 7^x‖10x is read as the product 10*x: for a power, write 10^x

1) Let's solve in

ℝ

x^{3} - 7 x^{2} + 10 x < 0

You can write the expression:

x^{3} - 7 x^{2} + 10 x

in factored form:

x \cdot (x^{2} - 7 x + 10)

Solution found:

x < 0

2) Let's solve in

ℝ

x^{2} - 7 x + 10 < 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

Δ = {(-7)}^{2} - 4 \cdot 1 \cdot 10

Δ = 49 - 4 \cdot 10

Δ = 49 - 40

Δ = 9

Δ > 0

, the equation has two solutions:

For the first solution, we obtain:

\frac{- b - \sqrt{Δ}}{2 a}

x1 = \frac{7 - \sqrt{9}}{2}

x1 = \frac{7 - 3}{2}

x1 = \frac{4}{2}

x1 = 2

The second solution is:

\frac{- b + \sqrt{Δ}}{2 a}

x2 = \frac{7 + \sqrt{9}}{2}

x2 = \frac{7 + 3}{2}

x2 = \frac{10}{2}

x2 = 5

Solution obtained:

x \in (2, 5)

Let's aggregate the different solutions to find the overall solution:

▶Solution obtained:

x \in (-∞, 0) \cup (2, 5)