Solve for m (m-4)(3m-10)=3(m-2)+18

$(m - 4) (3 m - 10) = 3 (m - 2) + 18$

Simplify $(m - 4) (3 m - 10)$ .

Rewrite.

$0 + 0 + (m - 4) (3 m - 10) = 3 (m - 2) + 18$

Simplify by adding zeros.

$(m - 4) (3 m - 10) = 3 (m - 2) + 18$

Expand $(m - 4) (3 m - 10)$ using the FOIL Method.

Apply the distributive property.

$m (3 m - 10) - 4 (3 m - 10) = 3 (m - 2) + 18$

Apply the distributive property.

$m (3 m) + m \cdot - 10 - 4 (3 m - 10) = 3 (m - 2) + 18$

Apply the distributive property.

$m (3 m) + m \cdot - 10 - 4 (3 m) - 4 \cdot - 10 = 3 (m - 2) + 18$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 m \cdot m + m \cdot - 10 - 4 (3 m) - 4 \cdot - 10 = 3 (m - 2) + 18$

Multiply $m$ by $m$ by adding the exponents.

Move $m$ .

$3 (m \cdot m) + m \cdot - 10 - 4 (3 m) - 4 \cdot - 10 = 3 (m - 2) + 18$

Multiply $m$ by $m$ .

$3 m^{2} + m \cdot - 10 - 4 (3 m) - 4 \cdot - 10 = 3 (m - 2) + 18$

Move $- 10$ to the left of $m$ .

$3 m^{2} - 10 \cdot m - 4 (3 m) - 4 \cdot - 10 = 3 (m - 2) + 18$

Multiply $3$ by $- 4$ .

$3 m^{2} - 10 m - 12 m - 4 \cdot - 10 = 3 (m - 2) + 18$

Multiply $- 4$ by $- 10$ .

$3 m^{2} - 10 m - 12 m + 40 = 3 (m - 2) + 18$

Subtract $12 m$ from $- 10 m$ .

$3 m^{2} - 22 m + 40 = 3 (m - 2) + 18$

Simplify $3 (m - 2) + 18$ .

Simplify each term.

Apply the distributive property.

$3 m^{2} - 22 m + 40 = 3 m + 3 \cdot - 2 + 18$

Multiply $3$ by $- 2$ .

$3 m^{2} - 22 m + 40 = 3 m - 6 + 18$

Add $- 6$ and $18$ .

$3 m^{2} - 22 m + 40 = 3 m + 12$

Move all terms containing $m$ to the left side of the equation.

Subtract $3 m$ from both sides of the equation.

$3 m^{2} - 22 m + 40 - 3 m = 12$

Subtract $3 m$ from $- 22 m$ .

$3 m^{2} - 25 m + 40 = 12$

Subtract $12$ from both sides of the equation.

$3 m^{2} - 25 m + 40 - 12 = 0$

Subtract $12$ from $40$ .

$3 m^{2} - 25 m + 28 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 28 = 84$ and whose sum is $b = - 25$ .

Factor $- 25$ out of $- 25 m$ .

$3 m^{2} - 25 m + 28 = 0$

Rewrite $- 25$ as $- 4$ plus $- 21$

$3 m^{2} + (- 4 - 21) m + 28 = 0$

Apply the distributive property.

$3 m^{2} - 4 m - 21 m + 28 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 m^{2} - 4 m) - 21 m + 28 = 0$

Factor out the greatest common factor (GCF) from each group.

$m (3 m - 4) - 7 (3 m - 4) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 m - 4$ .

$(3 m - 4) (m - 7) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 m - 4 = 0$

$m - 7 = 0$

Set $3 m - 4$ equal to $0$ and solve for $m$ .

Set $3 m - 4$ equal to $0$ .

$3 m - 4 = 0$

Solve $3 m - 4 = 0$ for $m$ .

Add $4$ to both sides of the equation.

$3 m = 4$

Divide each term in $3 m = 4$ by $3$ and simplify.

Divide each term in $3 m = 4$ by $3$ .

$\frac{3 m}{3} = \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 m}{3} = \frac{4}{3}$

Divide $m$ by $1$ .

$m = \frac{4}{3}$

Set $m - 7$ equal to $0$ and solve for $m$ .

Set $m - 7$ equal to $0$ .

$m - 7 = 0$

Add $7$ to both sides of the equation.

$m = 7$

The final solution is all the values that make $(3 m - 4) (m - 7) = 0$ true.

$m = \frac{4}{3}, 7$

The result can be shown in multiple forms.

Exact Form:

$m = \frac{4}{3}, 7$

Decimal Form:

$m = 1.\overline{3}, 7$

Mixed Number Form:

$m = 1 \frac{1}{3}, 7$

Solve for m (m-4)(3m-10)=3(m-2)+18

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