Algebra Examples

$\sqrt{- 5 x - 1} + \sqrt{5 x + 3} = 2$

Step 1

Subtract $\sqrt{5 x + 3}$ from both sides of the equation.

$\sqrt{- 5 x - 1} = 2 - \sqrt{5 x + 3}$

Step 2

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

${\sqrt{- 5 x - 1}}^{2} = {(2 - \sqrt{5 x + 3})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${({(- 5 x - 1)}^{\frac{1}{2}})}^{2} = {(2 - \sqrt{5 x + 3})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(- 5 x - 1)}^{\frac{1}{2} \cdot 2} = {(2 - \sqrt{5 x + 3})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(- 5 x - 1)}^{\frac{1}{2} \cdot 2} = {(2 - \sqrt{5 x + 3})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(- 5 x - 1)}^{1} = {(2 - \sqrt{5 x + 3})}^{2}$

Step 3.2.1.2

Simplify.

$- 5 x - 1 = {(2 - \sqrt{5 x + 3})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(2 - \sqrt{5 x + 3})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(2 - \sqrt{5 x + 3})}^{2}$ as $(2 - \sqrt{5 x + 3}) (2 - \sqrt{5 x + 3})$ .

$- 5 x - 1 = (2 - \sqrt{5 x + 3}) (2 - \sqrt{5 x + 3})$

Step 3.3.1.2

Expand $(2 - \sqrt{5 x + 3}) (2 - \sqrt{5 x + 3})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$- 5 x - 1 = 2 (2 - \sqrt{5 x + 3}) - \sqrt{5 x + 3} (2 - \sqrt{5 x + 3})$

Step 3.3.1.2.2

Apply the distributive property.

$- 5 x - 1 = 2 \cdot 2 + 2 (- \sqrt{5 x + 3}) - \sqrt{5 x + 3} (2 - \sqrt{5 x + 3})$

Step 3.3.1.2.3

Apply the distributive property.

$- 5 x - 1 = 2 \cdot 2 + 2 (- \sqrt{5 x + 3}) - \sqrt{5 x + 3} \cdot 2 - \sqrt{5 x + 3} (- \sqrt{5 x + 3})$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $2$ by $2$ .

$- 5 x - 1 = 4 + 2 (- \sqrt{5 x + 3}) - \sqrt{5 x + 3} \cdot 2 - \sqrt{5 x + 3} (- \sqrt{5 x + 3})$

Step 3.3.1.3.1.2

Multiply $- 1$ by $2$ .

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - \sqrt{5 x + 3} \cdot 2 - \sqrt{5 x + 3} (- \sqrt{5 x + 3})$

Step 3.3.1.3.1.3

Multiply $2$ by $- 1$ .

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} - \sqrt{5 x + 3} (- \sqrt{5 x + 3})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{5 x + 3} (- \sqrt{5 x + 3})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + 1 \sqrt{5 x + 3} \sqrt{5 x + 3}$

Step 3.3.1.3.1.4.2

Multiply $\sqrt{5 x + 3}$ by $1$ .

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + \sqrt{5 x + 3} \sqrt{5 x + 3}$

Step 3.3.1.3.1.4.3

Raise $\sqrt{5 x + 3}$ to the power of $1$ .

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {\sqrt{5 x + 3}}^{1} \sqrt{5 x + 3}$

Step 3.3.1.3.1.4.4

Raise $\sqrt{5 x + 3}$ to the power of $1$ .

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {\sqrt{5 x + 3}}^{1} {\sqrt{5 x + 3}}^{1}$

Step 3.3.1.3.1.4.5

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

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {\sqrt{5 x + 3}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {\sqrt{5 x + 3}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{5 x + 3}}^{2}$ as $5 x + 3$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 3}$ as ${(5 x + 3)}^{\frac{1}{2}}$ .

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {({(5 x + 3)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {(5 x + 3)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

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

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {(5 x + 3)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {(5 x + 3)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + {(5 x + 3)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$- 5 x - 1 = 4 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + 5 x + 3$

Step 3.3.1.3.2

Add $4$ and $3$ .

$- 5 x - 1 = 7 - 2 \sqrt{5 x + 3} - 2 \sqrt{5 x + 3} + 5 x$

Step 3.3.1.3.3

Subtract $2 \sqrt{5 x + 3}$ from $- 2 \sqrt{5 x + 3}$ .

$- 5 x - 1 = 7 - 4 \sqrt{5 x + 3} + 5 x$

Step 4

Solve for $- 4 \sqrt{5 x + 3}$ .

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Step 4.1

Rewrite the equation as $7 - 4 \sqrt{5 x + 3} + 5 x = - 5 x - 1$ .

$7 - 4 \sqrt{5 x + 3} + 5 x = - 5 x - 1$

Step 4.2

Move all terms not containing $- 4 \sqrt{5 x + 3}$ to the right side of the equation.

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Step 4.2.1

Subtract $7$ from both sides of the equation.

$- 4 \sqrt{5 x + 3} + 5 x = - 5 x - 1 - 7$

Step 4.2.2

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

$- 4 \sqrt{5 x + 3} = - 5 x - 1 - 7 - 5 x$

Step 4.2.3

Subtract $5 x$ from $- 5 x$ .

$- 4 \sqrt{5 x + 3} = - 10 x - 1 - 7$

Step 4.2.4

Subtract $7$ from $- 1$ .

$- 4 \sqrt{5 x + 3} = - 10 x - 8$

Step 5

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

${(- 4 \sqrt{5 x + 3})}^{2} = {(- 10 x - 8)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 3}$ as ${(5 x + 3)}^{\frac{1}{2}}$ .

${(- 4 {(5 x + 3)}^{\frac{1}{2}})}^{2} = {(- 10 x - 8)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 4 {(5 x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 4 {(5 x + 3)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(5 x + 3)}^{\frac{1}{2}})}^{2} = {(- 10 x - 8)}^{2}$

Step 6.2.1.2

Raise $- 4$ to the power of $2$ .

$16 {({(5 x + 3)}^{\frac{1}{2}})}^{2} = {(- 10 x - 8)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(5 x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$16 {(5 x + 3)}^{\frac{1}{2} \cdot 2} = {(- 10 x - 8)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$16 {(5 x + 3)}^{\frac{1}{2} \cdot 2} = {(- 10 x - 8)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 {(5 x + 3)}^{1} = {(- 10 x - 8)}^{2}$

Step 6.2.1.4

Simplify.

$16 (5 x + 3) = {(- 10 x - 8)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$16 (5 x) + 16 \cdot 3 = {(- 10 x - 8)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $5$ by $16$ .

$80 x + 16 \cdot 3 = {(- 10 x - 8)}^{2}$

Step 6.2.1.6.2

Multiply $16$ by $3$ .

$80 x + 48 = {(- 10 x - 8)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(- 10 x - 8)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(- 10 x - 8)}^{2}$ as $(- 10 x - 8) (- 10 x - 8)$ .

$80 x + 48 = (- 10 x - 8) (- 10 x - 8)$

Step 6.3.1.2

Expand $(- 10 x - 8) (- 10 x - 8)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$80 x + 48 = - 10 x (- 10 x - 8) - 8 (- 10 x - 8)$

Step 6.3.1.2.2

Apply the distributive property.

$80 x + 48 = - 10 x (- 10 x) - 10 x \cdot - 8 - 8 (- 10 x - 8)$

Step 6.3.1.2.3

Apply the distributive property.

$80 x + 48 = - 10 x (- 10 x) - 10 x \cdot - 8 - 8 (- 10 x) - 8 \cdot - 8$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$80 x + 48 = - 10 \cdot - 10 x \cdot x - 10 x \cdot - 8 - 8 (- 10 x) - 8 \cdot - 8$

Step 6.3.1.3.1.2

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

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Step 6.3.1.3.1.2.1

Move $x$ .

$80 x + 48 = - 10 \cdot - 10 (x \cdot x) - 10 x \cdot - 8 - 8 (- 10 x) - 8 \cdot - 8$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$80 x + 48 = - 10 \cdot - 10 x^{2} - 10 x \cdot - 8 - 8 (- 10 x) - 8 \cdot - 8$

Step 6.3.1.3.1.3

Multiply $- 10$ by $- 10$ .

$80 x + 48 = 100 x^{2} - 10 x \cdot - 8 - 8 (- 10 x) - 8 \cdot - 8$

Step 6.3.1.3.1.4

Multiply $- 8$ by $- 10$ .

$80 x + 48 = 100 x^{2} + 80 x - 8 (- 10 x) - 8 \cdot - 8$

Step 6.3.1.3.1.5

Multiply $- 10$ by $- 8$ .

$80 x + 48 = 100 x^{2} + 80 x + 80 x - 8 \cdot - 8$

Step 6.3.1.3.1.6

Multiply $- 8$ by $- 8$ .

$80 x + 48 = 100 x^{2} + 80 x + 80 x + 64$

Step 6.3.1.3.2

Add $80 x$ and $80 x$ .

$80 x + 48 = 100 x^{2} + 160 x + 64$

Step 7

Solve for $x$ .

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Step 7.1

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

$100 x^{2} + 160 x + 64 = 80 x + 48$

Step 7.2

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

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Step 7.2.1

Subtract $80 x$ from both sides of the equation.

$100 x^{2} + 160 x + 64 - 80 x = 48$

Step 7.2.2

Subtract $80 x$ from $160 x$ .

$100 x^{2} + 80 x + 64 = 48$

Step 7.3

Subtract $48$ from both sides of the equation.

$100 x^{2} + 80 x + 64 - 48 = 0$

Step 7.4

Subtract $48$ from $64$ .

$100 x^{2} + 80 x + 16 = 0$

Step 7.5

Factor the left side of the equation.

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Step 7.5.1

Factor $4$ out of $100 x^{2} + 80 x + 16$ .

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Step 7.5.1.1

Factor $4$ out of $100 x^{2}$ .

$4 (25 x^{2}) + 80 x + 16 = 0$

Step 7.5.1.2

Factor $4$ out of $80 x$ .

$4 (25 x^{2}) + 4 (20 x) + 16 = 0$

Step 7.5.1.3

Factor $4$ out of $16$ .

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

Step 7.5.1.4

Factor $4$ out of $4 (25 x^{2}) + 4 (20 x)$ .

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

Step 7.5.1.5

Factor $4$ out of $4 (25 x^{2} + 20 x) + 4 (4)$ .

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

Step 7.5.2

Factor using the perfect square rule.

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Step 7.5.2.1

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$4 ({(5 x)}^{2} + 20 x + 4) = 0$

Step 7.5.2.2

Rewrite $4$ as $2^{2}$ .

$4 ({(5 x)}^{2} + 20 x + 2^{2}) = 0$

Step 7.5.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$20 x = 2 \cdot (5 x) \cdot 2$

Step 7.5.2.4

Rewrite the polynomial.

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

Step 7.5.2.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 5 x$ and $b = 2$ .

$4 {(5 x + 2)}^{2} = 0$

Step 7.6

Divide each term in $4 {(5 x + 2)}^{2} = 0$ by $4$ and simplify.

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Step 7.6.1

Divide each term in $4 {(5 x + 2)}^{2} = 0$ by $4$ .

$\frac{4 {(5 x + 2)}^{2}}{4} = \frac{0}{4}$

Step 7.6.2

Simplify the left side.

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Step 7.6.2.1

Cancel the common factor of $4$ .

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Step 7.6.2.1.1

Cancel the common factor.

$\frac{4 {(5 x + 2)}^{2}}{4} = \frac{0}{4}$

Step 7.6.2.1.2

Divide ${(5 x + 2)}^{2}$ by $1$ .

${(5 x + 2)}^{2} = \frac{0}{4}$

Step 7.6.3

Simplify the right side.

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Step 7.6.3.1

Divide $0$ by $4$ .

${(5 x + 2)}^{2} = 0$

Step 7.7

Set the $5 x + 2$ equal to $0$ .

$5 x + 2 = 0$

Step 7.8

Solve for $x$ .

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Step 7.8.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 7.8.2

Divide each term in $5 x = - 2$ by $5$ and simplify.

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Step 7.8.2.1

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 7.8.2.2

Simplify the left side.

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Step 7.8.2.2.1

Cancel the common factor of $5$ .

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Step 7.8.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 7.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Step 7.8.2.3

Simplify the right side.

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Step 7.8.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{5}$

Decimal Form:

$x = - 0.4$