Solve x² + 3x = 0
Quadratic equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −3, x = 0 |
Step-by-step solution
6 steps-
1 Givenx^{2} + 3 x = 0
Solve for x.
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2 This is a quadratic in standard form ax² + bx + c = 0
Every quadratic equation can be arranged as ax² + bx + c = 0. Reading off a = 1, b = 3 and c = 0, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = 3,\quad c = 0 -
3 Every term contains x, so factor it out
Every term contains x, so x comes out as a common factor. Dividing both sides by x instead would be a mistake: it throws away the solution x = 0.
x \left(x + 3\right) = 0 -
4 Zero product property: a product is 0 only when one of its factors is 0
0 is the only number with this property: if a·b = 0, then a = 0 or b = 0. That is why the equation was first rearranged to … = 0 and factored: now each factor can be set to 0 on its own, giving a simpler equation for each.
x = 0\quad \text{or} \quad x + 3 = 0 -
5 Subtract 3 from both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
x = -3Move the constant terms to the right.
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6 Check\begin{aligned}x = -3:\quad 0 = 0\quad\checkmark\\ x = 0:\quad 0 = 0\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −3 | 0 | 0 | ✓ |
| 0 | 0 | 0 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.