Precalculus Examples

$\sqrt{x} + \sqrt{2 x} = 1$

Step 1

Subtract $\sqrt{2 x}$ from both sides of the equation.

$\sqrt{x} = 1 - \sqrt{2 x}$

Step 2

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

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

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite ${(1 - \sqrt{2 x})}^{2}$ as $(1 - \sqrt{2 x}) (1 - \sqrt{2 x})$ .

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

Step 3.3.1.2

Expand $(1 - \sqrt{2 x}) (1 - \sqrt{2 x})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 3.3.1.3.1.2

Multiply $- \sqrt{2 x}$ by $1$ .

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

Step 3.3.1.3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.3.1.3.1.4

Multiply $- \sqrt{2 x} (- \sqrt{2 x})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.3.1.4.2

Multiply $\sqrt{2 x}$ by $1$ .

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

Step 3.3.1.3.1.4.3

Raise $\sqrt{2 x}$ to the power of $1$ .

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

Step 3.3.1.3.1.4.4

Raise $\sqrt{2 x}$ to the power of $1$ .

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

Step 3.3.1.3.1.4.5

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

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

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

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

Step 3.3.1.3.1.5

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

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Step 3.3.1.3.1.5.1

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

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

Step 3.3.1.3.1.5.2

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

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

Step 3.3.1.3.1.5.3

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

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

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.

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

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.5.5

Simplify.

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

Step 3.3.1.3.2

Subtract $\sqrt{2 x}$ from $- \sqrt{2 x}$ .

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

Step 4

Solve for $- 2 \sqrt{2 x}$ .

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Step 4.1

Rewrite the equation as $1 - 2 \sqrt{2 x} + 2 x = x$ .

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

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

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

Step 4.2.2

Subtract $2 x$ from both sides of the equation.

$- 2 \sqrt{2 x} = x - 1 - 2 x$

Step 4.2.3

Subtract $2 x$ from $x$ .

$- 2 \sqrt{2 x} = - x - 1$

Step 5

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

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

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 $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Apply the product rule to $2 x$ .

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

Step 6.2.1.2

Multiply $- 2 \cdot 2^{\frac{1}{2}}$ .

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Step 6.2.1.2.1

Factor out negative.

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

Step 6.2.1.2.2

Raise $2$ to the power of $1$ .

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

Step 6.2.1.2.3

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

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

Step 6.2.1.2.4

Write $1$ as a fraction with a common denominator.

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

Step 6.2.1.2.5

Combine the numerators over the common denominator.

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

Step 6.2.1.2.6

Add $2$ and $1$ .

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

Step 6.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.2.1.3.1

Apply the product rule to $- 2^{\frac{3}{2}} x^{\frac{1}{2}}$ .

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

Step 6.2.1.3.2

Apply the product rule to $- 2^{\frac{3}{2}}$ .

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

Step 6.2.1.4

Raise $- 1$ to the power of $2$ .

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

Step 6.2.1.5

Multiply ${(2^{\frac{3}{2}})}^{2}$ by $1$ .

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

Step 6.2.1.6

Multiply the exponents in ${(2^{\frac{3}{2}})}^{2}$ .

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Step 6.2.1.6.1

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

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

Step 6.2.1.6.2

Cancel the common factor of $2$ .

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Step 6.2.1.6.2.1

Cancel the common factor.

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

Step 6.2.1.6.2.2

Rewrite the expression.

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

Step 6.2.1.7

Raise $2$ to the power of $3$ .

$8 {(x^{\frac{1}{2}})}^{2} = {(- x - 1)}^{2}$

Step 6.2.1.8

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

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Step 6.2.1.8.1

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

$8 x^{\frac{1}{2} \cdot 2} = {(- x - 1)}^{2}$

Step 6.2.1.8.2

Cancel the common factor of $2$ .

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Step 6.2.1.8.2.1

Cancel the common factor.

$8 x^{\frac{1}{2} \cdot 2} = {(- x - 1)}^{2}$

Step 6.2.1.8.2.2

Rewrite the expression.

$8 x^{1} = {(- x - 1)}^{2}$

Step 6.2.1.9

Simplify.

$8 x = {(- x - 1)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

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

$8 x = (- x - 1) (- x - 1)$

Step 6.3.1.2

Expand $(- x - 1) (- x - 1)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$8 x = - x (- x - 1) - 1 (- x - 1)$

Step 6.3.1.2.2

Apply the distributive property.

$8 x = - x (- x) - x \cdot - 1 - 1 (- x - 1)$

Step 6.3.1.2.3

Apply the distributive property.

$8 x = - x (- x) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

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.

$8 x = - 1 \cdot - 1 x \cdot x - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

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 $x$ .

$8 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$8 x = - 1 \cdot - 1 x^{2} - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Step 6.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$8 x = 1 x^{2} - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Step 6.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

$8 x = x^{2} - x \cdot - 1 - 1 (- x) - 1 \cdot - 1$

Step 6.3.1.3.1.5

Multiply $- x \cdot - 1$ .

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Step 6.3.1.3.1.5.1

Multiply $- 1$ by $- 1$ .

$8 x = x^{2} + 1 x - 1 (- x) - 1 \cdot - 1$

Step 6.3.1.3.1.5.2

Multiply $x$ by $1$ .

$8 x = x^{2} + x - 1 (- x) - 1 \cdot - 1$

Step 6.3.1.3.1.6

Multiply $- 1 (- x)$ .

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Step 6.3.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$8 x = x^{2} + x + 1 x - 1 \cdot - 1$

Step 6.3.1.3.1.6.2

Multiply $x$ by $1$ .

$8 x = x^{2} + x + x - 1 \cdot - 1$

Step 6.3.1.3.1.7

Multiply $- 1$ by $- 1$ .

$8 x = x^{2} + x + x + 1$

Step 6.3.1.3.2

Add $x$ and $x$ .

$8 x = x^{2} + 2 x + 1$

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.

$x^{2} + 2 x + 1 = 8 x$

Step 7.2

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

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Step 7.2.1

Subtract $8 x$ from both sides of the equation.

$x^{2} + 2 x + 1 - 8 x = 0$

Step 7.2.2

Subtract $8 x$ from $2 x$ .

$x^{2} - 6 x + 1 = 0$

Step 7.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.4

Substitute the values $a = 1$ , $b = - 6$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 7.5

Simplify.

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Step 7.5.1

Simplify the numerator.

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Step 7.5.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 7.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 7.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1}}{2 \cdot 1}$

Step 7.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4}}{2 \cdot 1}$

Step 7.5.1.3

Subtract $4$ from $36$ .

$x = \frac{6 \pm \sqrt{32}}{2 \cdot 1}$

Step 7.5.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 7.5.1.4.1

Factor $16$ out of $32$ .

$x = \frac{6 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 7.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 7.5.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 7.5.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 4 \sqrt{2}}{2}$

Step 7.5.3

Simplify $\frac{6 \pm 4 \sqrt{2}}{2}$ .

$x = 3 \pm 2 \sqrt{2}$

Step 7.6

The final answer is the combination of both solutions.

$x = 3 + 2 \sqrt{2}, 3 - 2 \sqrt{2}$

Step 8

Exclude the solutions that do not make $\sqrt{x} + \sqrt{2 x} = 1$ true.

$x = 3 - 2 \sqrt{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 3 - 2 \sqrt{2}$

Decimal Form:

$x = 0.17157287 \dots$