Solve x² − 4 = 3x
Quadratic equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = −1, x = 4 |
Step-by-step solution
9 steps-
1 Givenx^{2} - 4 = 3 x
Solve for x.
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2 Move every term to the left side so the right side is 0
Factoring and the quadratic formula both work on an equation of the form … = 0. Subtracting the right side from both sides gets there without changing the solutions.
x^{2} - 3 x - 4 = 0 -
3 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 = -4, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = -3,\quad c = -4 -
4 Factor the trinomial: find two numbers whose product is c = -4 and whose sum is b = -3
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 = -4 and add up to b = -3.
List the pairs of numbers whose product is -4, with their signs, and pick the pair whose sum is -3.
(-4) \cdot 1 = -4,\qquad (-4) + 1 = -3 -
5 Write the factored form\left(x - 4\right) \left(x + 1\right) = 0
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6 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 - 4 = 0\quad \text{or} \quad x + 1 = 0 -
7 Add 4 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 = 4Move the constant terms to the right.
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8 Subtract 1 from both sidesx = -1
Move the constant terms to the right.
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9 Check\begin{aligned}x = -1:\quad -3 = -3\quad\checkmark\\ x = 4:\quad 12 = 12\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| −1 | −3 | −3 | ✓ |
| 4 | 12 | 12 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.