Algebra Examples

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

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.

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Step 2.1

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

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

Step 2.2

Divide $2$ by $2$ .

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

Step 2.3

Simplify the left side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $2$ by $1$ .

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

Step 2.4

Simplify the right side.

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Step 2.4.1

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

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Step 2.4.1.1

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

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

Step 2.4.1.2

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

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Step 2.4.1.2.1

Apply the distributive property.

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

Step 2.4.1.2.2

Apply the distributive property.

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

Step 2.4.1.2.3

Apply the distributive property.

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

Step 2.4.1.3

Simplify and combine like terms.

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Step 2.4.1.3.1

Simplify each term.

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Step 2.4.1.3.1.1

Multiply $7$ by $7$ .

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

Step 2.4.1.3.1.2

Multiply $- 2$ by $7$ .

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

Step 2.4.1.3.1.3

Multiply $7$ by $- 2$ .

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

Step 2.4.1.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.4.1.3.1.5

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

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Step 2.4.1.3.1.5.1

Move $x$ .

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

Step 2.4.1.3.1.5.2

Multiply $x$ by $x$ .

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

Step 2.4.1.3.1.6

Multiply $- 2$ by $- 2$ .

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

Step 2.4.1.3.2

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

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

Step 3

Solve for $x$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Rewrite.

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

Step 3.2.2

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

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

Step 3.2.3

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

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Step 3.2.3.1

Apply the distributive property.

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

Step 3.2.3.2

Apply the distributive property.

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

Step 3.2.3.3

Apply the distributive property.

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

Step 3.2.4

Simplify and combine like terms.

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Step 3.2.4.1

Simplify each term.

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Step 3.2.4.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.4.1.2

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

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Step 3.2.4.1.2.1

Move $x$ .

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

Step 3.2.4.1.2.2

Multiply $x$ by $x$ .

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

Step 3.2.4.1.3

Multiply $2$ by $2$ .

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

Step 3.2.4.1.4

Multiply $- 7$ by $2$ .

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

Step 3.2.4.1.5

Multiply $2$ by $- 7$ .

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

Step 3.2.4.1.6

Multiply $- 7$ by $- 7$ .

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

Step 3.2.4.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

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

$49 - 28 x + 28 x = 49$

Step 3.3.3.3

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

$49 + 0 = 49$

Step 3.3.3.4

Add $49$ and $0$ .

$49 = 49$

Step 3.4

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:

$(- \infty, \infty)$