Solve x³ + 4x² − 25x − 100 = 0

Polynomial equation, worked out line by line the way a teacher would write it.

Answer

Solutions	x = −5, x = −4, x = 5

9 steps

1 Given

x^{3} + 4 x^{2} - 25 x - 100 = 0

Solve for x.
2 Rational root theorem: any rational root is ±(divisor of 100)/(divisor of 1)

x \in \{ -1,\; 1,\; -2,\; 2,\; -4,\; 4,\; -5,\; 5,\; -10,\; 10,\; -20,\; 20,\; -25,\; 25,\; -50,\; 50,\; \dots \}
3 Test the candidates: x = -4 makes the polynomial 0, so (x + 4) is a factor

4 \left(-4\right)^{2} - 100 - 25 \left(-4\right) + \left(-4\right)^{3} = 0
4 Divide the polynomial by (x + 4) (synthetic division)

\left(x + 4\right) \left(x^{2} - 25\right) = 0
5 This is a quadratic in standard form ax² + bx + c = 0

a = 1,\quad b = 0,\quad c = -25
6 There is no x term, so isolate x²

x^{2} = 25
7 Take the square root of both sides — remember both signs

x = \pm 5
8 Solutions

x = 5\quad \text{or} \quad x = -5
9 Check

\begin{aligned}x = -5:\quad 0 = 0\quad\checkmark\\ x = -4:\quad 0 = 0\quad\checkmark\\ x = 5:\quad 0 = 0\quad\checkmark\end{aligned}

Substituting each solution back makes both sides equal.

Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.