Algebra Examples

$\sqrt{4 x^{2} - 20 x + 25} = 5 - 2 x$

Step 1

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

${\sqrt{4 x^{2} - 20 x + 25}}^{2} = {(5 - 2 x)}^{2}$

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 $\sqrt{4 x^{2} - 20 x + 25}$ as ${(4 x^{2} - 20 x + 25)}^{\frac{1}{2}}$ .

${({(4 x^{2} - 20 x + 25)}^{\frac{1}{2}})}^{2} = {(5 - 2 x)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(4 x^{2} - 20 x + 25)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(4 x^{2} - 20 x + 25)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(4 x^{2} - 20 x + 25)}^{\frac{1}{2} \cdot 2} = {(5 - 2 x)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(4 x^{2} - 20 x + 25)}^{\frac{1}{2} \cdot 2} = {(5 - 2 x)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(4 x^{2} - 20 x + 25)}^{1} = {(5 - 2 x)}^{2}$

Step 2.2.1.2

Simplify.

$4 x^{2} - 20 x + 25 = {(5 - 2 x)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$4 x^{2} - 20 x + 25 = (5 - 2 x) (5 - 2 x)$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

$4 x^{2} - 20 x + 25 = 5 (5 - 2 x) - 2 x (5 - 2 x)$

Step 2.3.1.2.2

Apply the distributive property.

$4 x^{2} - 20 x + 25 = 5 \cdot 5 + 5 (- 2 x) - 2 x (5 - 2 x)$

Step 2.3.1.2.3

Apply the distributive property.

$4 x^{2} - 20 x + 25 = 5 \cdot 5 + 5 (- 2 x) - 2 x \cdot 5 - 2 x (- 2 x)$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $5$ by $5$ .

$4 x^{2} - 20 x + 25 = 25 + 5 (- 2 x) - 2 x \cdot 5 - 2 x (- 2 x)$

Step 2.3.1.3.1.2

Multiply $- 2$ by $5$ .

$4 x^{2} - 20 x + 25 = 25 - 10 x - 2 x \cdot 5 - 2 x (- 2 x)$

Step 2.3.1.3.1.3

Multiply $5$ by $- 2$ .

$4 x^{2} - 20 x + 25 = 25 - 10 x - 10 x - 2 x (- 2 x)$

Step 2.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 x^{2} - 20 x + 25 = 25 - 10 x - 10 x - 2 \cdot - 2 x \cdot x$

Step 2.3.1.3.1.5

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

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Step 2.3.1.3.1.5.1

Move $x$ .

$4 x^{2} - 20 x + 25 = 25 - 10 x - 10 x - 2 \cdot - 2 (x \cdot x)$

Step 2.3.1.3.1.5.2

Multiply $x$ by $x$ .

$4 x^{2} - 20 x + 25 = 25 - 10 x - 10 x - 2 \cdot - 2 x^{2}$

Step 2.3.1.3.1.6

Multiply $- 2$ by $- 2$ .

$4 x^{2} - 20 x + 25 = 25 - 10 x - 10 x + 4 x^{2}$

Step 2.3.1.3.2

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

$4 x^{2} - 20 x + 25 = 25 - 20 x + 4 x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Add $20 x$ to both sides of the equation.

$4 x^{2} - 20 x + 25 + 20 x = 25 + 4 x^{2}$

Step 3.1.2

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

$4 x^{2} - 20 x + 25 + 20 x - 4 x^{2} = 25$

Step 3.1.3

Combine the opposite terms in $4 x^{2} - 20 x + 25 + 20 x - 4 x^{2}$ .

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Step 3.1.3.1

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

$4 x^{2} + 0 + 25 - 4 x^{2} = 25$

Step 3.1.3.2

Add $4 x^{2}$ and $0$ .

$4 x^{2} + 25 - 4 x^{2} = 25$

Step 3.1.3.3

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

$0 + 25 = 25$

Step 3.1.3.4

Add $0$ and $25$ .

$25 = 25$

Step 3.2

Since $25 = 25$ , 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:

$(- \infty, \infty)$