Solve for z 7z^2+8z+3=3z^2

$7 z^{2} + 8 z + 3 = 3 z^{2}$

Step 1

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

Subtract $3 z^{2}$ from both sides of the equation.

$7 z^{2} + 8 z + 3 - 3 z^{2} = 0$

Subtract $3 z^{2}$ from $7 z^{2}$ .

$4 z^{2} + 8 z + 3 = 0$

Step 2

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 $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = 8$ .

Factor $8$ out of $8 z$ .

$4 z^{2} + 8 (z) + 3 = 0$

Rewrite $8$ as $2$ plus $6$

$4 z^{2} + (2 + 6) z + 3 = 0$

Apply the distributive property.

$4 z^{2} + 2 z + 6 z + 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 z^{2} + 2 z) + 6 z + 3 = 0$

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

$2 z (2 z + 1) + 3 (2 z + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 z + 1$ .

$(2 z + 1) (2 z + 3) = 0$

Step 3

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

$2 z + 1 = 0$

$2 z + 3 = 0$

Step 4

Set $2 z + 1$ equal to $0$ and solve for $z$ .

Set $2 z + 1$ equal to $0$ .

$2 z + 1 = 0$

Solve $2 z + 1 = 0$ for $z$ .

Subtract $1$ from both sides of the equation.

$2 z = - 1$

Divide each term in $2 z = - 1$ by $2$ and simplify.

Divide each term in $2 z = - 1$ by $2$ .

$\frac{2 z}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} = \frac{- 1}{2}$

Divide $z$ by $1$ .

$z = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$z = - \frac{1}{2}$

Step 5

Set $2 z + 3$ equal to $0$ and solve for $z$ .

Set $2 z + 3$ equal to $0$ .

$2 z + 3 = 0$

Solve $2 z + 3 = 0$ for $z$ .

Subtract $3$ from both sides of the equation.

$2 z = - 3$

Divide each term in $2 z = - 3$ by $2$ and simplify.

Divide each term in $2 z = - 3$ by $2$ .

$\frac{2 z}{2} = \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 z}{2} = \frac{- 3}{2}$

Divide $z$ by $1$ .

$z = \frac{- 3}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$z = - \frac{3}{2}$

Step 6

The final solution is all the values that make $(2 z + 1) (2 z + 3) = 0$ true.

$z = - \frac{1}{2}, - \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = - \frac{1}{2}, - \frac{3}{2}$

Decimal Form:

$z = - 0.5, - 1.5$

Mixed Number Form:

$z = - \frac{1}{2}, - 1 \frac{1}{2}$

Solve for z 7z^2+8z+3=3z^2

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