Solve x² − 4x − 5 = 0
Quadratic equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −1, x = 5 |
Step-by-step solution
8 steps-
1 Givenx^{2} - 4 x - 5 = 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 = -4 and c = -5, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = -4,\quad c = -5 -
3 Factor the trinomial: find two numbers whose product is c = -5 and whose sum is b = -4
The goal is to write x² + bx + c as (x + p)(x + q). Multiplying that out gives x² + (p + q)x + p·q, so p and q must multiply to c = -5 and add up to b = -4.
List the pairs of numbers whose product is -5, with their signs, and pick the pair whose sum is -4.
(-5) \cdot 1 = -5,\qquad (-5) + 1 = -4 -
4 Write the factored form\left(x - 5\right) \left(x + 1\right) = 0
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5 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 - 5 = 0\quad \text{or} \quad x + 1 = 0 -
6 Add 5 to 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 = 5Move the constant terms to the right.
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7 Subtract 1 from both sidesx = -1
Move the constant terms to the right.
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8 Check\begin{aligned}x = -1:\quad 0 = 0\quad\checkmark\\ x = 5:\quad 0 = 0\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −1 | 0 | 0 | ✓ |
| 5 | 0 | 0 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.