Solve for l l(3l-7)=20

$l (3 l - 7) = 20$

Step 1

Simplify $l (3 l - 7)$ .

Simplify by multiplying through.

Apply the distributive property.

$l (3 l) + l \cdot - 7 = 20$

Reorder.

Rewrite using the commutative property of multiplication.

$3 l \cdot l + l \cdot - 7 = 20$

Move $- 7$ to the left of $l$ .

$3 l \cdot l - 7 \cdot l = 20$

Multiply $l$ by $l$ by adding the exponents.

Move $l$ .

$3 (l \cdot l) - 7 \cdot l = 20$

Multiply $l$ by $l$ .

$3 l^{2} - 7 \cdot l = 20$

$3 l^{2} - 7 l = 20$

Step 2

Subtract $20$ from both sides of the equation.

$3 l^{2} - 7 l - 20 = 0$

Step 3

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 $a \cdot c = 3 \cdot - 20 = - 60$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 l$ .

$3 l^{2} - 7 l - 20 = 0$

Rewrite $- 7$ as $5$ plus $- 12$

$3 l^{2} + (5 - 12) l - 20 = 0$

Apply the distributive property.

$3 l^{2} + 5 l - 12 l - 20 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 l^{2} + 5 l) - 12 l - 20 = 0$

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

$l (3 l + 5) - 4 (3 l + 5) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 l + 5$ .

$(3 l + 5) (l - 4) = 0$

Step 4

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

$3 l + 5 = 0$

$l - 4 = 0$

Step 5

Set $3 l + 5$ equal to $0$ and solve for $l$ .

Set $3 l + 5$ equal to $0$ .

$3 l + 5 = 0$

Solve $3 l + 5 = 0$ for $l$ .

Subtract $5$ from both sides of the equation.

$3 l = - 5$

Divide each term in $3 l = - 5$ by $3$ and simplify.

Divide each term in $3 l = - 5$ by $3$ .

$\frac{3 l}{3} = \frac{- 5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 l}{3} = \frac{- 5}{3}$

Divide $l$ by $1$ .

$l = \frac{- 5}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$l = - \frac{5}{3}$

Step 6

Set $l - 4$ equal to $0$ and solve for $l$ .

Set $l - 4$ equal to $0$ .

$l - 4 = 0$

Add $4$ to both sides of the equation.

$l = 4$

Step 7

The final solution is all the values that make $(3 l + 5) (l - 4) = 0$ true.

$l = - \frac{5}{3}, 4$

Step 8

The result can be shown in multiple forms.

Exact Form:

$l = - \frac{5}{3}, 4$

Decimal Form:

$l = - 1.\overline{6}, 4$

Mixed Number Form:

$l = - 1 \frac{2}{3}, 4$

Solve for l l(3l-7)=20

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