Algebra Examples

$0.25 x + 1 = \sqrt{- x} + 4$

Step 1

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

$\sqrt{- x} + 4 = 0.25 x + 1$

Step 2

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

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Step 2.1

Subtract $4$ from both sides of the equation.

$\sqrt{- x} = 0.25 x + 1 - 4$

Step 2.2

Subtract $4$ from $1$ .

$\sqrt{- x} = 0.25 x - 3$

Step 3

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

${\sqrt{- x}}^{2} = {(0.25 x - 3)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${({(- x)}^{\frac{1}{2}})}^{2} = {(0.25 x - 3)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

${(- x)}^{\frac{1}{2} \cdot 2} = {(0.25 x - 3)}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(- x)}^{\frac{1}{2} \cdot 2} = {(0.25 x - 3)}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(- x)}^{1} = {(0.25 x - 3)}^{2}$

Step 4.2.1.2

Simplify.

$- x = {(0.25 x - 3)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(0.25 x - 3)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(0.25 x - 3)}^{2}$ as $(0.25 x - 3) (0.25 x - 3)$ .

$- x = (0.25 x - 3) (0.25 x - 3)$

Step 4.3.1.2

Expand $(0.25 x - 3) (0.25 x - 3)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$- x = 0.25 x (0.25 x - 3) - 3 (0.25 x - 3)$

Step 4.3.1.2.2

Apply the distributive property.

$- x = 0.25 x (0.25 x) + 0.25 x \cdot - 3 - 3 (0.25 x - 3)$

Step 4.3.1.2.3

Apply the distributive property.

$- x = 0.25 x (0.25 x) + 0.25 x \cdot - 3 - 3 (0.25 x) - 3 \cdot - 3$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$- x = 0.25 \cdot 0.25 x \cdot x + 0.25 x \cdot - 3 - 3 (0.25 x) - 3 \cdot - 3$

Step 4.3.1.3.1.2

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

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Step 4.3.1.3.1.2.1

Move $x$ .

$- x = 0.25 \cdot 0.25 (x \cdot x) + 0.25 x \cdot - 3 - 3 (0.25 x) - 3 \cdot - 3$

Step 4.3.1.3.1.2.2

Multiply $x$ by $x$ .

$- x = 0.25 \cdot 0.25 x^{2} + 0.25 x \cdot - 3 - 3 (0.25 x) - 3 \cdot - 3$

Step 4.3.1.3.1.3

Multiply $0.25$ by $0.25$ .

$- x = 0.0625 x^{2} + 0.25 x \cdot - 3 - 3 (0.25 x) - 3 \cdot - 3$

Step 4.3.1.3.1.4

Multiply $- 3$ by $0.25$ .

$- x = 0.0625 x^{2} - 0.75 x - 3 (0.25 x) - 3 \cdot - 3$

Step 4.3.1.3.1.5

Multiply $0.25$ by $- 3$ .

$- x = 0.0625 x^{2} - 0.75 x - 0.75 x - 3 \cdot - 3$

Step 4.3.1.3.1.6

Multiply $- 3$ by $- 3$ .

$- x = 0.0625 x^{2} - 0.75 x - 0.75 x + 9$

Step 4.3.1.3.2

Subtract $0.75 x$ from $- 0.75 x$ .

$- x = 0.0625 x^{2} - 1.5 x + 9$

Step 5

Solve for $x$ .

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Step 5.1

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

$0.0625 x^{2} - 1.5 x + 9 = - x$

Step 5.2

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

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Step 5.2.1

Add $x$ to both sides of the equation.

$0.0625 x^{2} - 1.5 x + 9 + x = 0$

Step 5.2.2

Add $- 1.5 x$ and $x$ .

$0.0625 x^{2} - 0.5 x + 9 = 0$

Step 5.3

Factor $0.0625$ out of $0.0625 x^{2} - 0.5 x + 9$ .

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Step 5.3.1

Factor $0.0625$ out of $0.0625 x^{2}$ .

$0.0625 (x^{2}) - 0.5 x + 9 = 0$

Step 5.3.2

Factor $0.0625$ out of $- 0.5 x$ .

$0.0625 (x^{2}) + 0.0625 (- 8 x) + 9 = 0$

Step 5.3.3

Factor $0.0625$ out of $9$ .

$0.0625 x^{2} + 0.0625 (- 8 x) + 0.0625 \cdot 144 = 0$

Step 5.3.4

Factor $0.0625$ out of $0.0625 x^{2} + 0.0625 (- 8 x)$ .

$0.0625 (x^{2} - 8 x) + 0.0625 \cdot 144 = 0$

Step 5.3.5

Factor $0.0625$ out of $0.0625 (x^{2} - 8 x) + 0.0625 \cdot 144$ .

$0.0625 (x^{2} - 8 x + 144) = 0$

Step 5.4

Divide each term in $0.0625 (x^{2} - 8 x + 144) = 0$ by $0.0625$ and simplify.

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Step 5.4.1

Divide each term in $0.0625 (x^{2} - 8 x + 144) = 0$ by $0.0625$ .

$\frac{0.0625 (x^{2} - 8 x + 144)}{0.0625} = \frac{0}{0.0625}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $0.0625$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{0.0625 (x^{2} - 8 x + 144)}{0.0625} = \frac{0}{0.0625}$

Step 5.4.2.1.2

Divide $x^{2} - 8 x + 144$ by $1$ .

$x^{2} - 8 x + 144 = \frac{0}{0.0625}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $0$ by $0.0625$ .

$x^{2} - 8 x + 144 = 0$

Step 5.5

Use the quadratic formula to find the solutions.

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

Step 5.6

Substitute the values $a = 1$ , $b = - 8$ , and $c = 144$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 144)}}{2 \cdot 1}$

Step 5.7

Simplify.

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Step 5.7.1

Simplify the numerator.

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Step 5.7.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 144}}{2 \cdot 1}$

Step 5.7.1.2

Multiply $- 4 \cdot 1 \cdot 144$ .

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Step 5.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 144}}{2 \cdot 1}$

Step 5.7.1.2.2

Multiply $- 4$ by $144$ .

$x = \frac{8 \pm \sqrt{64 - 576}}{2 \cdot 1}$

Step 5.7.1.3

Subtract $576$ from $64$ .

$x = \frac{8 \pm \sqrt{- 512}}{2 \cdot 1}$

Step 5.7.1.4

Rewrite $- 512$ as $- 1 (512)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 512}}{2 \cdot 1}$

Step 5.7.1.5

Rewrite $\sqrt{- 1 (512)}$ as $\sqrt{- 1} \cdot \sqrt{512}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{512}}{2 \cdot 1}$

Step 5.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{8 \pm i \cdot \sqrt{512}}{2 \cdot 1}$

Step 5.7.1.7

Rewrite $512$ as $16^{2} \cdot 2$ .

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Step 5.7.1.7.1

Factor $256$ out of $512$ .

$x = \frac{8 \pm i \cdot \sqrt{256 (2)}}{2 \cdot 1}$

Step 5.7.1.7.2

Rewrite $256$ as $16^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{16^{2} \cdot 2}}{2 \cdot 1}$

Step 5.7.1.8

Pull terms out from under the radical.

$x = \frac{8 \pm i \cdot (16 \sqrt{2})}{2 \cdot 1}$

Step 5.7.1.9

Move $16$ to the left of $i$ .

$x = \frac{8 \pm 16 i \sqrt{2}}{2 \cdot 1}$

Step 5.7.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 16 i \sqrt{2}}{2}$

Step 5.7.3

Simplify $\frac{8 \pm 16 i \sqrt{2}}{2}$ .

$x = 4 \pm 8 i \sqrt{2}$

Step 5.8

The final answer is the combination of both solutions.

$x = 4 + 8 i \sqrt{2}, 4 - 8 i \sqrt{2}$