Solve 6x² − 21x + 18 = 0
Quadratic equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = 3/2, x = 2 |
Step-by-step solution
12 steps-
1 Given6 x^{2} - 21 x + 18 = 0
Solve for x.
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2 Divide both sides by the common factor 3
Every coefficient is divisible by 3. Dividing both sides by 3 gives smaller numbers and the same solutions, because 0 divided by 3 is still 0.
2 x^{2} - 7 x + 6 = 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 = 2, b = -7 and c = 6, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 2,\quad b = -7,\quad c = 6 -
4 Split the middle term (ac method): find two numbers whose product is ac = 12 and whose sum is b = -7
When the x² term has a coefficient a other than 1, look for two numbers whose product is a·c = 12 and whose sum is b = -7. They split the middle term into two pieces, and the four terms can then be factored in pairs.
(-4) \cdot (-3) = 12,\qquad (-4) + (-3) = -7 -
5 Rewrite bx as the sum of those two terms
The two numbers add up to b, so writing the middle term as the sum of two terms does not change the expression. It just gives four terms that group in pairs.
2 x^{2} - 4 x - 3 x + 6 = 0 -
6 Factor each pair by grouping
Take the greatest common factor out of the first two terms and out of the last two. Because of how the middle term was split, the same bracket shows up in both pieces.
2 x \left(x - 2\right) + 3 \left(2 - x\right) = 0 -
7 Factor out the common bracket
Both pieces contain the same bracket, so it factors out like any common factor: A·B + C·B = (A + C)·B.
\left(x - 2\right) \left(2 x - 3\right) = 0 -
8 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 - 2 = 0\quad \text{or} \quad 2 x - 3 = 0 -
9 Add 2 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 = 2Move the constant terms to the right.
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10 Add 3 to both sides2 x = 3
Move the constant terms to the right.
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11 Divide both sides by 2
Dividing both sides by 2 undoes the multiplication by 2, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
x = \frac{3}{2} -
12 Check\begin{aligned}x = \frac{3}{2}:\quad 0 = 0\quad\checkmark\\ x = 2:\quad 0 = 0\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 3/2 | 0 | 0 | ✓ |
| 2 | 0 | 0 | ✓ |
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