Precalculus Examples

$\sqrt{10 a - 1} = 1 + \sqrt{5 a - 1}$

Step 1

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

${\sqrt{10 a - 1}}^{2} = {(1 + \sqrt{5 a - 1})}^{2}$

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 $\sqrt{10 a - 1}$ as ${(10 a - 1)}^{\frac{1}{2}}$ .

${({(10 a - 1)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{5 a - 1})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(10 a - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(10 a - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(10 a - 1)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{5 a - 1})}^{2}$

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.

${(10 a - 1)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{5 a - 1})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(10 a - 1)}^{1} = {(1 + \sqrt{5 a - 1})}^{2}$

Step 2.2.1.2

Simplify.

$10 a - 1 = {(1 + \sqrt{5 a - 1})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(1 + \sqrt{5 a - 1})}^{2}$ .

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Step 2.3.1.1

Rewrite ${(1 + \sqrt{5 a - 1})}^{2}$ as $(1 + \sqrt{5 a - 1}) (1 + \sqrt{5 a - 1})$ .

$10 a - 1 = (1 + \sqrt{5 a - 1}) (1 + \sqrt{5 a - 1})$

Step 2.3.1.2

Expand $(1 + \sqrt{5 a - 1}) (1 + \sqrt{5 a - 1})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$10 a - 1 = 1 (1 + \sqrt{5 a - 1}) + \sqrt{5 a - 1} (1 + \sqrt{5 a - 1})$

Step 2.3.1.2.2

Apply the distributive property.

$10 a - 1 = 1 \cdot 1 + 1 \sqrt{5 a - 1} + \sqrt{5 a - 1} (1 + \sqrt{5 a - 1})$

Step 2.3.1.2.3

Apply the distributive property.

$10 a - 1 = 1 \cdot 1 + 1 \sqrt{5 a - 1} + \sqrt{5 a - 1} \cdot 1 + \sqrt{5 a - 1} \sqrt{5 a - 1}$

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 $1$ by $1$ .

$10 a - 1 = 1 + 1 \sqrt{5 a - 1} + \sqrt{5 a - 1} \cdot 1 + \sqrt{5 a - 1} \sqrt{5 a - 1}$

Step 2.3.1.3.1.2

Multiply $\sqrt{5 a - 1}$ by $1$ .

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} \cdot 1 + \sqrt{5 a - 1} \sqrt{5 a - 1}$

Step 2.3.1.3.1.3

Multiply $\sqrt{5 a - 1}$ by $1$ .

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + \sqrt{5 a - 1} \sqrt{5 a - 1}$

Step 2.3.1.3.1.4

Multiply $\sqrt{5 a - 1} \sqrt{5 a - 1}$ .

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Step 2.3.1.3.1.4.1

Raise $\sqrt{5 a - 1}$ to the power of $1$ .

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {\sqrt{5 a - 1}}^{1} \sqrt{5 a - 1}$

Step 2.3.1.3.1.4.2

Raise $\sqrt{5 a - 1}$ to the power of $1$ .

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {\sqrt{5 a - 1}}^{1} {\sqrt{5 a - 1}}^{1}$

Step 2.3.1.3.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {\sqrt{5 a - 1}}^{1 + 1}$

Step 2.3.1.3.1.4.4

Add $1$ and $1$ .

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {\sqrt{5 a - 1}}^{2}$

Step 2.3.1.3.1.5

Rewrite ${\sqrt{5 a - 1}}^{2}$ as $5 a - 1$ .

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Step 2.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 a - 1}$ as ${(5 a - 1)}^{\frac{1}{2}}$ .

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {({(5 a - 1)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.3.1.5.2

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

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {(5 a - 1)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {(5 a - 1)}^{\frac{2}{2}}$

Step 2.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.5.4.1

Cancel the common factor.

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {(5 a - 1)}^{\frac{2}{2}}$

Step 2.3.1.3.1.5.4.2

Rewrite the expression.

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + {(5 a - 1)}^{1}$

Step 2.3.1.3.1.5.5

Simplify.

$10 a - 1 = 1 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + 5 a - 1$

Step 2.3.1.3.2

Subtract $1$ from $1$ .

$10 a - 1 = 0 + \sqrt{5 a - 1} + \sqrt{5 a - 1} + 5 a$

Step 2.3.1.3.3

Add $0$ and $\sqrt{5 a - 1}$ .

$10 a - 1 = \sqrt{5 a - 1} + \sqrt{5 a - 1} + 5 a$

Step 2.3.1.3.4

Add $\sqrt{5 a - 1}$ and $\sqrt{5 a - 1}$ .

$10 a - 1 = 2 \sqrt{5 a - 1} + 5 a$

Step 3

Solve for $2 \sqrt{5 a - 1}$ .

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Step 3.1

Rewrite the equation as $2 \sqrt{5 a - 1} + 5 a = 10 a - 1$ .

$2 \sqrt{5 a - 1} + 5 a = 10 a - 1$

Step 3.2

Move all terms not containing $2 \sqrt{5 a - 1}$ to the right side of the equation.

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Step 3.2.1

Subtract $5 a$ from both sides of the equation.

$2 \sqrt{5 a - 1} = 10 a - 1 - 5 a$

Step 3.2.2

Subtract $5 a$ from $10 a$ .

$2 \sqrt{5 a - 1} = 5 a - 1$

Step 4

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

${(2 \sqrt{5 a - 1})}^{2} = {(5 a - 1)}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 a - 1}$ as ${(5 a - 1)}^{\frac{1}{2}}$ .

${(2 {(5 a - 1)}^{\frac{1}{2}})}^{2} = {(5 a - 1)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(2 {(5 a - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $2 {(5 a - 1)}^{\frac{1}{2}}$ .

$2^{2} {({(5 a - 1)}^{\frac{1}{2}})}^{2} = {(5 a - 1)}^{2}$

Step 5.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(5 a - 1)}^{\frac{1}{2}})}^{2} = {(5 a - 1)}^{2}$

Step 5.2.1.3

Multiply the exponents in ${({(5 a - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.3.1

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

$4 {(5 a - 1)}^{\frac{1}{2} \cdot 2} = {(5 a - 1)}^{2}$

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

$4 {(5 a - 1)}^{\frac{1}{2} \cdot 2} = {(5 a - 1)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$4 {(5 a - 1)}^{1} = {(5 a - 1)}^{2}$

Step 5.2.1.4

Simplify.

$4 (5 a - 1) = {(5 a - 1)}^{2}$

Step 5.2.1.5

Apply the distributive property.

$4 (5 a) + 4 \cdot - 1 = {(5 a - 1)}^{2}$

Step 5.2.1.6

Multiply.

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Step 5.2.1.6.1

Multiply $5$ by $4$ .

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

Step 5.2.1.6.2

Multiply $4$ by $- 1$ .

$20 a - 4 = {(5 a - 1)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(5 a - 1)}^{2}$ .

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Step 5.3.1.1

Rewrite ${(5 a - 1)}^{2}$ as $(5 a - 1) (5 a - 1)$ .

$20 a - 4 = (5 a - 1) (5 a - 1)$

Step 5.3.1.2

Expand $(5 a - 1) (5 a - 1)$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$20 a - 4 = 5 a (5 a - 1) - 1 (5 a - 1)$

Step 5.3.1.2.2

Apply the distributive property.

$20 a - 4 = 5 a (5 a) + 5 a \cdot - 1 - 1 (5 a - 1)$

Step 5.3.1.2.3

Apply the distributive property.

$20 a - 4 = 5 a (5 a) + 5 a \cdot - 1 - 1 (5 a) - 1 \cdot - 1$

Step 5.3.1.3

Simplify and combine like terms.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$20 a - 4 = 5 \cdot 5 a \cdot a + 5 a \cdot - 1 - 1 (5 a) - 1 \cdot - 1$

Step 5.3.1.3.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 5.3.1.3.1.2.1

Move $a$ .

$20 a - 4 = 5 \cdot 5 (a \cdot a) + 5 a \cdot - 1 - 1 (5 a) - 1 \cdot - 1$

Step 5.3.1.3.1.2.2

Multiply $a$ by $a$ .

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

Step 5.3.1.3.1.3

Multiply $5$ by $5$ .

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

Step 5.3.1.3.1.4

Multiply $- 1$ by $5$ .

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

Step 5.3.1.3.1.5

Multiply $5$ by $- 1$ .

$20 a - 4 = 25 a^{2} - 5 a - 5 a - 1 \cdot - 1$

Step 5.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$20 a - 4 = 25 a^{2} - 5 a - 5 a + 1$

Step 5.3.1.3.2

Subtract $5 a$ from $- 5 a$ .

$20 a - 4 = 25 a^{2} - 10 a + 1$

Step 6

Solve for $a$ .

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Step 6.1

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

$25 a^{2} - 10 a + 1 = 20 a - 4$

Step 6.2

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

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Step 6.2.1

Subtract $20 a$ from both sides of the equation.

$25 a^{2} - 10 a + 1 - 20 a = - 4$

Step 6.2.2

Subtract $20 a$ from $- 10 a$ .

$25 a^{2} - 30 a + 1 = - 4$

Step 6.3

Add $4$ to both sides of the equation.

$25 a^{2} - 30 a + 1 + 4 = 0$

Step 6.4

Add $1$ and $4$ .

$25 a^{2} - 30 a + 5 = 0$

Step 6.5

Factor the left side of the equation.

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Step 6.5.1

Factor $5$ out of $25 a^{2} - 30 a + 5$ .

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Step 6.5.1.1

Factor $5$ out of $25 a^{2}$ .

$5 (5 a^{2}) - 30 a + 5 = 0$

Step 6.5.1.2

Factor $5$ out of $- 30 a$ .

$5 (5 a^{2}) + 5 (- 6 a) + 5 = 0$

Step 6.5.1.3

Factor $5$ out of $5$ .

$5 (5 a^{2}) + 5 (- 6 a) + 5 (1) = 0$

Step 6.5.1.4

Factor $5$ out of $5 (5 a^{2}) + 5 (- 6 a)$ .

$5 (5 a^{2} - 6 a) + 5 (1) = 0$

Step 6.5.1.5

Factor $5$ out of $5 (5 a^{2} - 6 a) + 5 (1)$ .

$5 (5 a^{2} - 6 a + 1) = 0$

Step 6.5.2

Factor.

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Step 6.5.2.1

Factor by grouping.

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Step 6.5.2.1.1

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 $a \cdot c = 5 \cdot 1 = 5$ and whose sum is $b = - 6$ .

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Step 6.5.2.1.1.1

Factor $- 6$ out of $- 6 a$ .

$5 (5 a^{2} - 6 a + 1) = 0$

Step 6.5.2.1.1.2

Rewrite $- 6$ as $- 1$ plus $- 5$

$5 (5 a^{2} + (- 1 - 5) a + 1) = 0$

Step 6.5.2.1.1.3

Apply the distributive property.

$5 (5 a^{2} - 1 a - 5 a + 1) = 0$

Step 6.5.2.1.2

Factor out the greatest common factor from each group.

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Step 6.5.2.1.2.1

Group the first two terms and the last two terms.

$5 ((5 a^{2} - 1 a) - 5 a + 1) = 0$

Step 6.5.2.1.2.2

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

$5 (a (5 a - 1) - (5 a - 1)) = 0$

Step 6.5.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 a - 1$ .

$5 ((5 a - 1) (a - 1)) = 0$

Step 6.5.2.2

Remove unnecessary parentheses.

$5 (5 a - 1) (a - 1) = 0$

Step 6.6

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

$5 a - 1 = 0$

$a - 1 = 0$

Step 6.7

Set $5 a - 1$ equal to $0$ and solve for $a$ .

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Step 6.7.1

Set $5 a - 1$ equal to $0$ .

$5 a - 1 = 0$

Step 6.7.2

Solve $5 a - 1 = 0$ for $a$ .

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Step 6.7.2.1

Add $1$ to both sides of the equation.

$5 a = 1$

Step 6.7.2.2

Divide each term in $5 a = 1$ by $5$ and simplify.

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Step 6.7.2.2.1

Divide each term in $5 a = 1$ by $5$ .

$\frac{5 a}{5} = \frac{1}{5}$

Step 6.7.2.2.2

Simplify the left side.

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Step 6.7.2.2.2.1

Cancel the common factor of $5$ .

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Step 6.7.2.2.2.1.1

Cancel the common factor.

$\frac{5 a}{5} = \frac{1}{5}$

Step 6.7.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{1}{5}$

Step 6.8

Set $a - 1$ equal to $0$ and solve for $a$ .

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Step 6.8.1

Set $a - 1$ equal to $0$ .

$a - 1 = 0$

Step 6.8.2

Add $1$ to both sides of the equation.

$a = 1$

Step 6.9

The final solution is all the values that make $5 (5 a - 1) (a - 1) = 0$ true.

$a = \frac{1}{5}, 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{1}{5}, 1$

Decimal Form:

$a = 0.2, 1$