Find the Roots (Zeros) 4x^2+77=-36x

$4 x^{2} + 77 = - 36 x$

Add $36 x$ to both sides of the equation.

$4 x^{2} + 77 + 36 x = 0$

Factor by grouping.

Reorder terms.

$4 x^{2} + 36 x + 77 = 0$

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 $a \cdot c = 4 \cdot 77 = 308$ and whose sum is $b = 36$ .

Factor $36$ out of $36 x$ .

$4 x^{2} + 36 (x) + 77 = 0$

Rewrite $36$ as $14$ plus $22$

$4 x^{2} + (14 + 22) x + 77 = 0$

Apply the distributive property.

$4 x^{2} + 14 x + 22 x + 77 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} + 14 x) + 22 x + 77 = 0$

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

$2 x (2 x + 7) + 11 (2 x + 7) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 7$ .

$(2 x + 7) (2 x + 11) = 0$

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

$2 x + 7 = 0$

$2 x + 11 = 0$

Set $2 x + 7$ equal to $0$ and solve for $x$ .

Set $2 x + 7$ equal to $0$ .

$2 x + 7 = 0$

Solve $2 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Divide each term in $2 x = - 7$ by $2$ and simplify.

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Set $2 x + 11$ equal to $0$ and solve for $x$ .

Set $2 x + 11$ equal to $0$ .

$2 x + 11 = 0$

Solve $2 x + 11 = 0$ for $x$ .

Subtract $11$ from both sides of the equation.

$2 x = - 11$

Divide each term in $2 x = - 11$ by $2$ and simplify.

Divide each term in $2 x = - 11$ by $2$ .

$\frac{2 x}{2} = \frac{- 11}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 11}{2}$

Divide $x$ by $1$ .

$x = \frac{- 11}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{11}{2}$

The final solution is all the values that make $(2 x + 7) (2 x + 11) = 0$ true.

$x = - \frac{7}{2}, - \frac{11}{2}$

Find the Roots (Zeros) 4x^2+77=-36x

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