Solve for x (x-3)^2-(2x+5)^2=-16

${(x - 3)}^{2} - {(2 x + 5)}^{2} = - 16$

Step 1

Add $16$ to both sides of the equation.

${(x - 3)}^{2} - {(2 x + 5)}^{2} + 16 = 0$

Step 2

Simplify ${(x - 3)}^{2} - {(2 x + 5)}^{2} + 16$ .

Simplify each term.

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

$(x - 3) (x - 3) - {(2 x + 5)}^{2} + 16 = 0$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$x (x - 3) - 3 (x - 3) - {(2 x + 5)}^{2} + 16 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 (x - 3) - {(2 x + 5)}^{2} + 16 = 0$

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - {(2 x + 5)}^{2} + 16 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 - {(2 x + 5)}^{2} + 16 = 0$

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 - {(2 x + 5)}^{2} + 16 = 0$

Multiply $- 3$ by $- 3$ .

$x^{2} - 3 x - 3 x + 9 - {(2 x + 5)}^{2} + 16 = 0$

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 6 x + 9 - {(2 x + 5)}^{2} + 16 = 0$

Rewrite ${(2 x + 5)}^{2}$ as $(2 x + 5) (2 x + 5)$ .

$x^{2} - 6 x + 9 - ((2 x + 5) (2 x + 5)) + 16 = 0$

Expand $(2 x + 5) (2 x + 5)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 6 x + 9 - (2 x (2 x + 5) + 5 (2 x + 5)) + 16 = 0$

Apply the distributive property.

$x^{2} - 6 x + 9 - (2 x (2 x) + 2 x \cdot 5 + 5 (2 x + 5)) + 16 = 0$

Apply the distributive property.

$x^{2} - 6 x + 9 - (2 x (2 x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x^{2} - 6 x + 9 - (2 \cdot 2 x \cdot x + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

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

Move $x$ .

$x^{2} - 6 x + 9 - (2 \cdot 2 (x \cdot x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Multiply $x$ by $x$ .

$x^{2} - 6 x + 9 - (2 \cdot 2 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Multiply $2$ by $2$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Multiply $5$ by $2$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 10 x + 5 (2 x) + 5 \cdot 5) + 16 = 0$

Multiply $2$ by $5$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 10 x + 10 x + 5 \cdot 5) + 16 = 0$

Multiply $5$ by $5$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 10 x + 10 x + 25) + 16 = 0$

Add $10 x$ and $10 x$ .

$x^{2} - 6 x + 9 - (4 x^{2} + 20 x + 25) + 16 = 0$

Apply the distributive property.

$x^{2} - 6 x + 9 - (4 x^{2}) - (20 x) - 1 \cdot 25 + 16 = 0$

Simplify.

Multiply $4$ by $- 1$ .

$x^{2} - 6 x + 9 - 4 x^{2} - (20 x) - 1 \cdot 25 + 16 = 0$

Multiply $20$ by $- 1$ .

$x^{2} - 6 x + 9 - 4 x^{2} - 20 x - 1 \cdot 25 + 16 = 0$

Multiply $- 1$ by $25$ .

$x^{2} - 6 x + 9 - 4 x^{2} - 20 x - 25 + 16 = 0$

Subtract $4 x^{2}$ from $x^{2}$ .

$- 3 x^{2} - 6 x + 9 - 20 x - 25 + 16 = 0$

Subtract $20 x$ from $- 6 x$ .

$- 3 x^{2} - 26 x + 9 - 25 + 16 = 0$

Subtract $25$ from $9$ .

$- 3 x^{2} - 26 x - 16 + 16 = 0$

Combine the opposite terms in $- 3 x^{2} - 26 x - 16 + 16$ .

Add $- 16$ and $16$ .

$- 3 x^{2} - 26 x + 0 = 0$

Add $- 3 x^{2} - 26 x$ and $0$ .

$- 3 x^{2} - 26 x = 0$

Step 3

Factor $- x$ out of $- 3 x^{2} - 26 x$ .

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

$- x (3 x) - 26 x = 0$

Factor $- x$ out of $- 26 x$ .

$- x (3 x) - x \cdot 26 = 0$

Factor $- x$ out of $- x (3 x) - x (26)$ .

$- x (3 x + 26) = 0$

Step 4

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

$x = 0$

$3 x + 26 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $3 x + 26$ equal to $0$ and solve for $x$ .

Set $3 x + 26$ equal to $0$ .

$3 x + 26 = 0$

Solve $3 x + 26 = 0$ for $x$ .

Subtract $26$ from both sides of the equation.

$3 x = - 26$

Divide each term in $3 x = - 26$ by $3$ and simplify.

Divide each term in $3 x = - 26$ by $3$ .

$\frac{3 x}{3} = \frac{- 26}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 26}{3}$

Divide $x$ by $1$ .

$x = \frac{- 26}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{26}{3}$

Step 7

The final solution is all the values that make $- x (3 x + 26) = 0$ true.

$x = 0, - \frac{26}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{26}{3}$

Decimal Form:

$x = 0, - 8.\overline{6}$

Mixed Number Form:

$x = 0, - 8 \frac{2}{3}$

Solve for x (x-3)^2-(2x+5)^2=-16

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