Pre-Algebra Examples

$\sqrt{5 x - 12} - \sqrt{x} - 3 = - 1$

Step 1

Move all terms not containing $\sqrt{5 x - 12}$ to the right side of the equation.

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Step 1.1

Add $\sqrt{x}$ to both sides of the equation.

$\sqrt{5 x - 12} - 3 = - 1 + \sqrt{x}$

Step 1.2

Add $3$ to both sides of the equation.

$\sqrt{5 x - 12} = - 1 + \sqrt{x} + 3$

Step 1.3

Add $- 1$ and $3$ .

$\sqrt{5 x - 12} = 2 + \sqrt{x}$

Step 2

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

${\sqrt{5 x - 12}}^{2} = {(2 + \sqrt{x})}^{2}$

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 $\sqrt{5 x - 12}$ as ${(5 x - 12)}^{\frac{1}{2}}$ .

${({(5 x - 12)}^{\frac{1}{2}})}^{2} = {(2 + \sqrt{x})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(5 x - 12)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(5 x - 12)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(5 x - 12)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{x})}^{2}$

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.

${(5 x - 12)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{x})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(5 x - 12)}^{1} = {(2 + \sqrt{x})}^{2}$

Step 3.2.1.2

Simplify.

$5 x - 12 = {(2 + \sqrt{x})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$5 x - 12 = (2 + \sqrt{x}) (2 + \sqrt{x})$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

$5 x - 12 = 2 (2 + \sqrt{x}) + \sqrt{x} (2 + \sqrt{x})$

Step 3.3.1.2.2

Apply the distributive property.

$5 x - 12 = 2 \cdot 2 + 2 \sqrt{x} + \sqrt{x} (2 + \sqrt{x})$

Step 3.3.1.2.3

Apply the distributive property.

$5 x - 12 = 2 \cdot 2 + 2 \sqrt{x} + \sqrt{x} \cdot 2 + \sqrt{x} \sqrt{x}$

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 $2$ by $2$ .

$5 x - 12 = 4 + 2 \sqrt{x} + \sqrt{x} \cdot 2 + \sqrt{x} \sqrt{x}$

Step 3.3.1.3.1.2

Move $2$ to the left of $\sqrt{x}$ .

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \cdot \sqrt{x} + \sqrt{x} \sqrt{x}$

Step 3.3.1.3.1.3

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 3.3.1.3.1.3.1

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

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

Step 3.3.1.3.1.3.2

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

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

Step 3.3.1.3.1.3.3

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

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + {\sqrt{x}}^{2}$

Step 3.3.1.3.1.4

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

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Step 3.3.1.3.1.4.1

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

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + x^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + x^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + x^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$5 x - 12 = 4 + 2 \sqrt{x} + 2 \sqrt{x} + x$

Step 3.3.1.3.2

Add $2 \sqrt{x}$ and $2 \sqrt{x}$ .

$5 x - 12 = 4 + 4 \sqrt{x} + x$

Step 4

Solve for $4 \sqrt{x}$ .

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Step 4.1

Rewrite the equation as $4 + 4 \sqrt{x} + x = 5 x - 12$ .

$4 + 4 \sqrt{x} + x = 5 x - 12$

Step 4.2

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

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$4 \sqrt{x} + x = 5 x - 12 - 4$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$4 \sqrt{x} = 5 x - 12 - 4 - x$

Step 4.2.3

Subtract $x$ from $5 x$ .

$4 \sqrt{x} = 4 x - 12 - 4$

Step 4.2.4

Subtract $4$ from $- 12$ .

$4 \sqrt{x} = 4 x - 16$

Step 5

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

${(4 \sqrt{x})}^{2} = {(4 x - 16)}^{2}$

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

${(4 x^{\frac{1}{2}})}^{2} = {(4 x - 16)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(4 x^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $4 x^{\frac{1}{2}}$ .

$4^{2} {(x^{\frac{1}{2}})}^{2} = {(4 x - 16)}^{2}$

Step 6.2.1.2

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

$16 {(x^{\frac{1}{2}})}^{2} = {(4 x - 16)}^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$16 x^{\frac{1}{2} \cdot 2} = {(4 x - 16)}^{2}$

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.

$16 x^{\frac{1}{2} \cdot 2} = {(4 x - 16)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 x^{1} = {(4 x - 16)}^{2}$

Step 6.2.1.4

Simplify.

$16 x = {(4 x - 16)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(4 x - 16)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(4 x - 16)}^{2}$ as $(4 x - 16) (4 x - 16)$ .

$16 x = (4 x - 16) (4 x - 16)$

Step 6.3.1.2

Expand $(4 x - 16) (4 x - 16)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$16 x = 4 x (4 x - 16) - 16 (4 x - 16)$

Step 6.3.1.2.2

Apply the distributive property.

$16 x = 4 x (4 x) + 4 x \cdot - 16 - 16 (4 x - 16)$

Step 6.3.1.2.3

Apply the distributive property.

$16 x = 4 x (4 x) + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

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.

$16 x = 4 \cdot 4 x \cdot x + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

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 $x$ .

$16 x = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$16 x = 4 \cdot 4 x^{2} + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

Step 6.3.1.3.1.3

Multiply $4$ by $4$ .

$16 x = 16 x^{2} + 4 x \cdot - 16 - 16 (4 x) - 16 \cdot - 16$

Step 6.3.1.3.1.4

Multiply $- 16$ by $4$ .

$16 x = 16 x^{2} - 64 x - 16 (4 x) - 16 \cdot - 16$

Step 6.3.1.3.1.5

Multiply $4$ by $- 16$ .

$16 x = 16 x^{2} - 64 x - 64 x - 16 \cdot - 16$

Step 6.3.1.3.1.6

Multiply $- 16$ by $- 16$ .

$16 x = 16 x^{2} - 64 x - 64 x + 256$

Step 6.3.1.3.2

Subtract $64 x$ from $- 64 x$ .

$16 x = 16 x^{2} - 128 x + 256$

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.

$16 x^{2} - 128 x + 256 = 16 x$

Step 7.2

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

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Step 7.2.1

Subtract $16 x$ from both sides of the equation.

$16 x^{2} - 128 x + 256 - 16 x = 0$

Step 7.2.2

Subtract $16 x$ from $- 128 x$ .

$16 x^{2} - 144 x + 256 = 0$

Step 7.3

Factor $16$ out of $16 x^{2} - 144 x + 256$ .

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Step 7.3.1

Factor $16$ out of $16 x^{2}$ .

$16 (x^{2}) - 144 x + 256 = 0$

Step 7.3.2

Factor $16$ out of $- 144 x$ .

$16 (x^{2}) + 16 (- 9 x) + 256 = 0$

Step 7.3.3

Factor $16$ out of $256$ .

$16 x^{2} + 16 (- 9 x) + 16 \cdot 16 = 0$

Step 7.3.4

Factor $16$ out of $16 x^{2} + 16 (- 9 x)$ .

$16 (x^{2} - 9 x) + 16 \cdot 16 = 0$

Step 7.3.5

Factor $16$ out of $16 (x^{2} - 9 x) + 16 \cdot 16$ .

$16 (x^{2} - 9 x + 16) = 0$

Step 7.4

Divide each term in $16 (x^{2} - 9 x + 16) = 0$ by $16$ and simplify.

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Step 7.4.1

Divide each term in $16 (x^{2} - 9 x + 16) = 0$ by $16$ .

$\frac{16 (x^{2} - 9 x + 16)}{16} = \frac{0}{16}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $16$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{16 (x^{2} - 9 x + 16)}{16} = \frac{0}{16}$

Step 7.4.2.1.2

Divide $x^{2} - 9 x + 16$ by $1$ .

$x^{2} - 9 x + 16 = \frac{0}{16}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Divide $0$ by $16$ .

$x^{2} - 9 x + 16 = 0$

Step 7.5

Use the quadratic formula to find the solutions.

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

Step 7.6

Substitute the values $a = 1$ , $b = - 9$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Step 7.7

Simplify.

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Step 7.7.1

Simplify the numerator.

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Step 7.7.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 7.7.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 7.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 16}}{2 \cdot 1}$

Step 7.7.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{9 \pm \sqrt{81 - 64}}{2 \cdot 1}$

Step 7.7.1.3

Subtract $64$ from $81$ .

$x = \frac{9 \pm \sqrt{17}}{2 \cdot 1}$

Step 7.7.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{17}}{2}$

Step 7.8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.8.1

Simplify the numerator.

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Step 7.8.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 7.8.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 7.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 16}}{2 \cdot 1}$

Step 7.8.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{9 \pm \sqrt{81 - 64}}{2 \cdot 1}$

Step 7.8.1.3

Subtract $64$ from $81$ .

$x = \frac{9 \pm \sqrt{17}}{2 \cdot 1}$

Step 7.8.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{17}}{2}$

Step 7.8.3

Change the $\pm$ to $+$ .

$x = \frac{9 + \sqrt{17}}{2}$

Step 7.9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.9.1

Simplify the numerator.

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Step 7.9.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 7.9.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 7.9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 16}}{2 \cdot 1}$

Step 7.9.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{9 \pm \sqrt{81 - 64}}{2 \cdot 1}$

Step 7.9.1.3

Subtract $64$ from $81$ .

$x = \frac{9 \pm \sqrt{17}}{2 \cdot 1}$

Step 7.9.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{17}}{2}$

Step 7.9.3

Change the $\pm$ to $-$ .

$x = \frac{9 - \sqrt{17}}{2}$

Step 7.10

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{17}}{2}, \frac{9 - \sqrt{17}}{2}$

Step 8

Exclude the solutions that do not make $\sqrt{5 x - 12} - \sqrt{x} - 3 = - 1$ true.

$x = \frac{9 + \sqrt{17}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9 + \sqrt{17}}{2}$

Decimal Form:

$x = 6.56155281 \dots$