Solve for x square root of (2x-7)^2=7-2x

$\sqrt{{(2 x - 7)}^{2}} = 7 - 2 x$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{{(2 x - 7)}^{2}}}^{2} = {(7 - 2 x)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(2 x - 7)}^{2}}$ as ${(2 x - 7)}^{\frac{2}{2}}$ .

${({(2 x - 7)}^{\frac{2}{2}})}^{2} = {(7 - 2 x)}^{2}$

Divide $2$ by $2$ .

${({(2 x - 7)}^{1})}^{2} = {(7 - 2 x)}^{2}$

Simplify the left side.

Multiply the exponents in ${({(2 x - 7)}^{1})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x - 7)}^{1 \cdot 2} = {(7 - 2 x)}^{2}$

Multiply $2$ by $1$ .

${(2 x - 7)}^{2} = {(7 - 2 x)}^{2}$

Simplify the right side.

Simplify ${(7 - 2 x)}^{2}$ .

Rewrite ${(7 - 2 x)}^{2}$ as $(7 - 2 x) (7 - 2 x)$ .

${(2 x - 7)}^{2} = (7 - 2 x) (7 - 2 x)$

Expand $(7 - 2 x) (7 - 2 x)$ using the FOIL Method.

Apply the distributive property.

${(2 x - 7)}^{2} = 7 (7 - 2 x) - 2 x (7 - 2 x)$

Apply the distributive property.

${(2 x - 7)}^{2} = 7 \cdot 7 + 7 (- 2 x) - 2 x (7 - 2 x)$

Apply the distributive property.

${(2 x - 7)}^{2} = 7 \cdot 7 + 7 (- 2 x) - 2 x \cdot 7 - 2 x (- 2 x)$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

${(2 x - 7)}^{2} = 49 + 7 (- 2 x) - 2 x \cdot 7 - 2 x (- 2 x)$

Multiply $- 2$ by $7$ .

${(2 x - 7)}^{2} = 49 - 14 x - 2 x \cdot 7 - 2 x (- 2 x)$

Multiply $7$ by $- 2$ .

${(2 x - 7)}^{2} = 49 - 14 x - 14 x - 2 x (- 2 x)$

Rewrite using the commutative property of multiplication.

${(2 x - 7)}^{2} = 49 - 14 x - 14 x - 2 \cdot - 2 x \cdot x$

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

Move $x$ .

${(2 x - 7)}^{2} = 49 - 14 x - 14 x - 2 \cdot - 2 (x \cdot x)$

Multiply $x$ by $x$ .

${(2 x - 7)}^{2} = 49 - 14 x - 14 x - 2 \cdot - 2 x^{2}$

Multiply $- 2$ by $- 2$ .

${(2 x - 7)}^{2} = 49 - 14 x - 14 x + 4 x^{2}$

Subtract $14 x$ from $- 14 x$ .

${(2 x - 7)}^{2} = 49 - 28 x + 4 x^{2}$

Step 3

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$49 - 28 x + 4 x^{2} = {(2 x - 7)}^{2}$

Simplify ${(2 x - 7)}^{2}$ .

Rewrite.

$49 - 28 x + 4 x^{2} = 0 + 0 + {(2 x - 7)}^{2}$

Rewrite ${(2 x - 7)}^{2}$ as $(2 x - 7) (2 x - 7)$ .

$49 - 28 x + 4 x^{2} = (2 x - 7) (2 x - 7)$

Expand $(2 x - 7) (2 x - 7)$ using the FOIL Method.

Apply the distributive property.

$49 - 28 x + 4 x^{2} = 2 x (2 x - 7) - 7 (2 x - 7)$

Apply the distributive property.

$49 - 28 x + 4 x^{2} = 2 x (2 x) + 2 x \cdot - 7 - 7 (2 x - 7)$

Apply the distributive property.

$49 - 28 x + 4 x^{2} = 2 x (2 x) + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$49 - 28 x + 4 x^{2} = 2 \cdot 2 x \cdot x + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

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

Move $x$ .

$49 - 28 x + 4 x^{2} = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

Multiply $x$ by $x$ .

$49 - 28 x + 4 x^{2} = 2 \cdot 2 x^{2} + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

Multiply $2$ by $2$ .

$49 - 28 x + 4 x^{2} = 4 x^{2} + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

Multiply $- 7$ by $2$ .

$49 - 28 x + 4 x^{2} = 4 x^{2} - 14 x - 7 (2 x) - 7 \cdot - 7$

Multiply $2$ by $- 7$ .

$49 - 28 x + 4 x^{2} = 4 x^{2} - 14 x - 14 x - 7 \cdot - 7$

Multiply $- 7$ by $- 7$ .

$49 - 28 x + 4 x^{2} = 4 x^{2} - 14 x - 14 x + 49$

Subtract $14 x$ from $- 14 x$ .

$49 - 28 x + 4 x^{2} = 4 x^{2} - 28 x + 49$

Move all terms containing $x$ to the left side of the equation.

Subtract $4 x^{2}$ from both sides of the equation.

$49 - 28 x + 4 x^{2} - 4 x^{2} = - 28 x + 49$

Add $28 x$ to both sides of the equation.

$49 - 28 x + 4 x^{2} - 4 x^{2} + 28 x = 49$

Combine the opposite terms in $49 - 28 x + 4 x^{2} - 4 x^{2} + 28 x$ .

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

$49 - 28 x + 0 + 28 x = 49$

Add $49 - 28 x$ and $0$ .

$49 - 28 x + 28 x = 49$

Add $- 28 x$ and $28 x$ .

$49 + 0 = 49$

Add $49$ and $0$ .

$49 = 49$

Since $49 = 49$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Solve for x square root of (2x-7)^2=7-2x

Related Questions

Support LCMGCF.com by disabling your adblocker.