Solve for x (2x+3)^2-14x-21=-6

${(2 x + 3)}^{2} - 14 x - 21 = - 6$

Step 1

Add $6$ to both sides of the equation.

${(2 x + 3)}^{2} - 14 x - 21 + 6 = 0$

Step 2

Simplify ${(2 x + 3)}^{2} - 14 x - 21 + 6$ .

Simplify each term.

Rewrite ${(2 x + 3)}^{2}$ as $(2 x + 3) (2 x + 3)$ .

$(2 x + 3) (2 x + 3) - 14 x - 21 + 6 = 0$

Expand $(2 x + 3) (2 x + 3)$ using the FOIL Method.

Apply the distributive property.

$2 x (2 x + 3) + 3 (2 x + 3) - 14 x - 21 + 6 = 0$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 3 + 3 (2 x + 3) - 14 x - 21 + 6 = 0$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - 14 x - 21 + 6 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - 14 x - 21 + 6 = 0$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - 14 x - 21 + 6 = 0$

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - 14 x - 21 + 6 = 0$

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3 - 14 x - 21 + 6 = 0$

Multiply $3$ by $2$ .

$4 x^{2} + 6 x + 3 (2 x) + 3 \cdot 3 - 14 x - 21 + 6 = 0$

Multiply $2$ by $3$ .

$4 x^{2} + 6 x + 6 x + 3 \cdot 3 - 14 x - 21 + 6 = 0$

Multiply $3$ by $3$ .

$4 x^{2} + 6 x + 6 x + 9 - 14 x - 21 + 6 = 0$

Add $6 x$ and $6 x$ .

$4 x^{2} + 12 x + 9 - 14 x - 21 + 6 = 0$

Subtract $14 x$ from $12 x$ .

$4 x^{2} - 2 x + 9 - 21 + 6 = 0$

Subtract $21$ from $9$ .

$4 x^{2} - 2 x - 12 + 6 = 0$

Add $- 12$ and $6$ .

$4 x^{2} - 2 x - 6 = 0$

Step 3

Factor the left side of the equation.

Factor $2$ out of $4 x^{2} - 2 x - 6$ .

Factor $2$ out of $4 x^{2}$ .

$2 (2 x^{2}) - 2 x - 6 = 0$

Factor $2$ out of $- 2 x$ .

$2 (2 x^{2}) + 2 (- x) - 6 = 0$

Factor $2$ out of $- 6$ .

$2 (2 x^{2}) + 2 (- x) + 2 (- 3) = 0$

Factor $2$ out of $2 (2 x^{2}) + 2 (- x)$ .

$2 (2 x^{2} - x) + 2 (- 3) = 0$

Factor $2$ out of $2 (2 x^{2} - x) + 2 (- 3)$ .

$2 (2 x^{2} - x - 3) = 0$

Factor.

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$2 (2 x^{2} - (x) - 3) = 0$

Rewrite $- 1$ as $2$ plus $- 3$

$2 (2 x^{2} + (2 - 3) x - 3) = 0$

Apply the distributive property.

$2 (2 x^{2} + 2 x - 3 x - 3) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((2 x^{2} + 2 x) - 3 x - 3) = 0$

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

$2 (2 x (x + 1) - 3 (x + 1)) = 0$

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

$2 ((x + 1) (2 x - 3)) = 0$

Remove unnecessary parentheses.

$2 (x + 1) (2 x - 3) = 0$

Step 4

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

$x + 1 = 0$

$2 x - 3 = 0$

Step 5

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

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

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1, 1.5$

Mixed Number Form:

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

Solve for x (2x+3)^2-14x-21=-6

Related Questions

Support LCMGCF.com by disabling your adblocker.