Solve for q 20q^2+19q+7=4

$20 q^{2} + 19 q + 7 = 4$

Step 1

Subtract $4$ from both sides of the equation.

$20 q^{2} + 19 q + 7 - 4 = 0$

Step 2

Subtract $4$ from $7$ .

$20 q^{2} + 19 q + 3 = 0$

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 $a \cdot c = 20 \cdot 3 = 60$ and whose sum is $b = 19$ .

Factor $19$ out of $19 q$ .

$20 q^{2} + 19 (q) + 3 = 0$

Rewrite $19$ as $4$ plus $15$

$20 q^{2} + (4 + 15) q + 3 = 0$

Apply the distributive property.

$20 q^{2} + 4 q + 15 q + 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(20 q^{2} + 4 q) + 15 q + 3 = 0$

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

$4 q (5 q + 1) + 3 (5 q + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 q + 1$ .

$(5 q + 1) (4 q + 3) = 0$

Step 4

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

$5 q + 1 = 0$

$4 q + 3 = 0$

Step 5

Set $5 q + 1$ equal to $0$ and solve for $q$ .

Set $5 q + 1$ equal to $0$ .

$5 q + 1 = 0$

Solve $5 q + 1 = 0$ for $q$ .

Subtract $1$ from both sides of the equation.

$5 q = - 1$

Divide each term in $5 q = - 1$ by $5$ and simplify.

Divide each term in $5 q = - 1$ by $5$ .

$\frac{5 q}{5} = \frac{- 1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 q}{5} = \frac{- 1}{5}$

Divide $q$ by $1$ .

$q = \frac{- 1}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$q = - \frac{1}{5}$

Step 6

Set $4 q + 3$ equal to $0$ and solve for $q$ .

Set $4 q + 3$ equal to $0$ .

$4 q + 3 = 0$

Solve $4 q + 3 = 0$ for $q$ .

Subtract $3$ from both sides of the equation.

$4 q = - 3$

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

Divide each term in $4 q = - 3$ by $4$ .

$\frac{4 q}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 q}{4} = \frac{- 3}{4}$

Divide $q$ by $1$ .

$q = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$q = - \frac{3}{4}$

Step 7

The final solution is all the values that make $(5 q + 1) (4 q + 3) = 0$ true.

$q = - \frac{1}{5}, - \frac{3}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$q = - \frac{1}{5}, - \frac{3}{4}$

Decimal Form:

$q = - 0.2, - 0.75$

Solve for q 20q^2+19q+7=4

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