Solve for x (2x-5)^2-(x-6)^2=80

${(2 x - 5)}^{2} - {(x - 6)}^{2} = 80$

Subtract $80$ from both sides of the equation.

${(2 x - 5)}^{2} - {(x - 6)}^{2} - 80 = 0$

Simplify ${(2 x - 5)}^{2} - {(x - 6)}^{2} - 80$ .

Simplify each term.

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

$(2 x - 5) (2 x - 5) - {(x - 6)}^{2} - 80 = 0$

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

Apply the distributive property.

$2 x (2 x - 5) - 5 (2 x - 5) - {(x - 6)}^{2} - 80 = 0$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5) - {(x - 6)}^{2} - 80 = 0$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(x - 6)}^{2} - 80 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(x - 6)}^{2} - 80 = 0$

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

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(x - 6)}^{2} - 80 = 0$

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(x - 6)}^{2} - 80 = 0$

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(x - 6)}^{2} - 80 = 0$

Multiply $- 5$ by $2$ .

$4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5 - {(x - 6)}^{2} - 80 = 0$

Multiply $2$ by $- 5$ .

$4 x^{2} - 10 x - 10 x - 5 \cdot - 5 - {(x - 6)}^{2} - 80 = 0$

Multiply $- 5$ by $- 5$ .

$4 x^{2} - 10 x - 10 x + 25 - {(x - 6)}^{2} - 80 = 0$

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

$4 x^{2} - 20 x + 25 - {(x - 6)}^{2} - 80 = 0$

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$4 x^{2} - 20 x + 25 - ((x - 6) (x - 6)) - 80 = 0$

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$4 x^{2} - 20 x + 25 - (x (x - 6) - 6 (x - 6)) - 80 = 0$

Apply the distributive property.

$4 x^{2} - 20 x + 25 - (x \cdot x + x \cdot - 6 - 6 (x - 6)) - 80 = 0$

Apply the distributive property.

$4 x^{2} - 20 x + 25 - (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) - 80 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$4 x^{2} - 20 x + 25 - (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) - 80 = 0$

Move $- 6$ to the left of $x$ .

$4 x^{2} - 20 x + 25 - (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) - 80 = 0$

Multiply $- 6$ by $- 6$ .

$4 x^{2} - 20 x + 25 - (x^{2} - 6 x - 6 x + 36) - 80 = 0$

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

$4 x^{2} - 20 x + 25 - (x^{2} - 12 x + 36) - 80 = 0$

Apply the distributive property.

$4 x^{2} - 20 x + 25 - x^{2} - (- 12 x) - 1 \cdot 36 - 80 = 0$

Simplify.

Multiply $- 12$ by $- 1$ .

$4 x^{2} - 20 x + 25 - x^{2} + 12 x - 1 \cdot 36 - 80 = 0$

Multiply $- 1$ by $36$ .

$4 x^{2} - 20 x + 25 - x^{2} + 12 x - 36 - 80 = 0$

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

$3 x^{2} - 20 x + 25 + 12 x - 36 - 80 = 0$

Add $- 20 x$ and $12 x$ .

$3 x^{2} - 8 x + 25 - 36 - 80 = 0$

Subtract $36$ from $25$ .

$3 x^{2} - 8 x - 11 - 80 = 0$

Subtract $80$ from $- 11$ .

$3 x^{2} - 8 x - 91 = 0$

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 $a \cdot c = 3 \cdot - 91 = - 273$ and whose sum is $b = - 8$ .

Factor $- 8$ out of $- 8 x$ .

$3 x^{2} - 8 x - 91 = 0$

Rewrite $- 8$ as $13$ plus $- 21$

$3 x^{2} + (13 - 21) x - 91 = 0$

Apply the distributive property.

$3 x^{2} + 13 x - 21 x - 91 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} + 13 x) - 21 x - 91 = 0$

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

$x (3 x + 13) - 7 (3 x + 13) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 13$ .

$(3 x + 13) (x - 7) = 0$

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

$3 x + 13 = 0$

$x - 7 = 0$

Set $3 x + 13$ equal to $0$ and solve for $x$ .

Set $3 x + 13$ equal to $0$ .

$3 x + 13 = 0$

Solve $3 x + 13 = 0$ for $x$ .

Subtract $13$ from both sides of the equation.

$3 x = - 13$

Divide each term in $3 x = - 13$ by $3$ and simplify.

Divide each term in $3 x = - 13$ by $3$ .

$\frac{3 x}{3} = \frac{- 13}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 13}{3}$

Divide $x$ by $1$ .

$x = \frac{- 13}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{13}{3}$

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.

$x = 7$

The final solution is all the values that make $(3 x + 13) (x - 7) = 0$ true.

$x = - \frac{13}{3}, 7$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{13}{3}, 7$

Decimal Form:

$x = - 4.\overline{3}, 7$

Mixed Number Form:

$x = - 4 \frac{1}{3}, 7$

Solve for x (2x-5)^2-(x-6)^2=80

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