Solve for x (2x-3)^2-(x+5)^2=-23

${(2 x - 3)}^{2} - {(x + 5)}^{2} = - 23$

Step 1

Add $23$ to both sides of the equation.

${(2 x - 3)}^{2} - {(x + 5)}^{2} + 23 = 0$

Step 2

Simplify ${(2 x - 3)}^{2} - {(x + 5)}^{2} + 23$ .

Simplify each term.

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

$(2 x - 3) (2 x - 3) - {(x + 5)}^{2} + 23 = 0$

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

Apply the distributive property.

$2 x (2 x - 3) - 3 (2 x - 3) - {(x + 5)}^{2} + 23 = 0$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 3 - 3 (2 x - 3) - {(x + 5)}^{2} + 23 = 0$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - {(x + 5)}^{2} + 23 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - {(x + 5)}^{2} + 23 = 0$

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

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - {(x + 5)}^{2} + 23 = 0$

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - {(x + 5)}^{2} + 23 = 0$

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3 - {(x + 5)}^{2} + 23 = 0$

Multiply $- 3$ by $2$ .

$4 x^{2} - 6 x - 3 (2 x) - 3 \cdot - 3 - {(x + 5)}^{2} + 23 = 0$

Multiply $2$ by $- 3$ .

$4 x^{2} - 6 x - 6 x - 3 \cdot - 3 - {(x + 5)}^{2} + 23 = 0$

Multiply $- 3$ by $- 3$ .

$4 x^{2} - 6 x - 6 x + 9 - {(x + 5)}^{2} + 23 = 0$

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

$4 x^{2} - 12 x + 9 - {(x + 5)}^{2} + 23 = 0$

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

$4 x^{2} - 12 x + 9 - ((x + 5) (x + 5)) + 23 = 0$

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

Apply the distributive property.

$4 x^{2} - 12 x + 9 - (x (x + 5) + 5 (x + 5)) + 23 = 0$

Apply the distributive property.

$4 x^{2} - 12 x + 9 - (x \cdot x + x \cdot 5 + 5 (x + 5)) + 23 = 0$

Apply the distributive property.

$4 x^{2} - 12 x + 9 - (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) + 23 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$4 x^{2} - 12 x + 9 - (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) + 23 = 0$

Move $5$ to the left of $x$ .

$4 x^{2} - 12 x + 9 - (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) + 23 = 0$

Multiply $5$ by $5$ .

$4 x^{2} - 12 x + 9 - (x^{2} + 5 x + 5 x + 25) + 23 = 0$

Add $5 x$ and $5 x$ .

$4 x^{2} - 12 x + 9 - (x^{2} + 10 x + 25) + 23 = 0$

Apply the distributive property.

$4 x^{2} - 12 x + 9 - x^{2} - (10 x) - 1 \cdot 25 + 23 = 0$

Simplify.

Multiply $10$ by $- 1$ .

$4 x^{2} - 12 x + 9 - x^{2} - 10 x - 1 \cdot 25 + 23 = 0$

Multiply $- 1$ by $25$ .

$4 x^{2} - 12 x + 9 - x^{2} - 10 x - 25 + 23 = 0$

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

$3 x^{2} - 12 x + 9 - 10 x - 25 + 23 = 0$

Subtract $10 x$ from $- 12 x$ .

$3 x^{2} - 22 x + 9 - 25 + 23 = 0$

Subtract $25$ from $9$ .

$3 x^{2} - 22 x - 16 + 23 = 0$

Add $- 16$ and $23$ .

$3 x^{2} - 22 x + 7 = 0$

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 $a \cdot c = 3 \cdot 7 = 21$ and whose sum is $b = - 22$ .

Factor $- 22$ out of $- 22 x$ .

$3 x^{2} - 22 x + 7 = 0$

Rewrite $- 22$ as $- 1$ plus $- 21$

$3 x^{2} + (- 1 - 21) x + 7 = 0$

Apply the distributive property.

$3 x^{2} - 1 x - 21 x + 7 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} - 1 x) - 21 x + 7 = 0$

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

$x (3 x - 1) - 7 (3 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$(3 x - 1) (x - 7) = 0$

Step 4

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

$3 x - 1 = 0$

$x - 7 = 0$

Step 5

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 6

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 7

The final solution is all the values that make $(3 x - 1) (x - 7) = 0$ true.

$x = \frac{1}{3}, 7$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}, 7$

Decimal Form:

$x = 0.\overline{3}, 7$

Solve for x (2x-3)^2-(x+5)^2=-23

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