Algebra Examples

$y = g \sqrt{x} + h$

Step 1

Rewrite the equation as $g \sqrt{x} + h = y$ .

$g \sqrt{x} + h = y$

Step 2

Subtract $h$ from both sides of the equation.

$g \sqrt{x} = y - h$

Step 3

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

${(g \sqrt{x})}^{2} = {(y - h)}^{2}$

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

Cancel the common factor of $2$ .

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Step 4.2.1.2.2.1

Cancel the common factor.

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

Step 4.2.1.2.2.2

Rewrite the expression.

$g^{2} x^{1} = {(y - h)}^{2}$

Step 4.2.1.3

Simplify.

$g^{2} x = {(y - h)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(y - h)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(y - h)}^{2}$ as $(y - h) (y - h)$ .

$g^{2} x = (y - h) (y - h)$

Step 4.3.1.2

Expand $(y - h) (y - h)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$g^{2} x = y (y - h) - h (y - h)$

Step 4.3.1.2.2

Apply the distributive property.

$g^{2} x = y \cdot y + y (- h) - h (y - h)$

Step 4.3.1.2.3

Apply the distributive property.

$g^{2} x = y \cdot y + y (- h) - h y - h (- h)$

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

Multiply $y$ by $y$ .

$g^{2} x = y^{2} + y (- h) - h y - h (- h)$

Step 4.3.1.3.1.2

Rewrite using the commutative property of multiplication.

$g^{2} x = y^{2} - y h - h y - h (- h)$

Step 4.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$g^{2} x = y^{2} - y h - h y - 1 \cdot - 1 h \cdot h$

Step 4.3.1.3.1.4

Multiply $h$ by $h$ by adding the exponents.

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Step 4.3.1.3.1.4.1

Move $h$ .

$g^{2} x = y^{2} - y h - h y - 1 \cdot - 1 (h \cdot h)$

Step 4.3.1.3.1.4.2

Multiply $h$ by $h$ .

$g^{2} x = y^{2} - y h - h y - 1 \cdot - 1 h^{2}$

Step 4.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$g^{2} x = y^{2} - y h - h y + 1 h^{2}$

Step 4.3.1.3.1.6

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

$g^{2} x = y^{2} - y h - h y + h^{2}$

Step 4.3.1.3.2

Subtract $h y$ from $- y h$ .

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Step 4.3.1.3.2.1

Move $y$ .

$g^{2} x = y^{2} - h y - h y + h^{2}$

Step 4.3.1.3.2.2

Subtract $h y$ from $- h y$ .

$g^{2} x = y^{2} - 2 h y + h^{2}$

Step 5

Divide each term in $g^{2} x = y^{2} - 2 h y + h^{2}$ by $g^{2}$ and simplify.

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Step 5.1

Divide each term in $g^{2} x = y^{2} - 2 h y + h^{2}$ by $g^{2}$ .

$\frac{g^{2} x}{g^{2}} = \frac{y^{2}}{g^{2}} + \frac{- 2 h y}{g^{2}} + \frac{h^{2}}{g^{2}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $g^{2}$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{g^{2} x}{g^{2}} = \frac{y^{2}}{g^{2}} + \frac{- 2 h y}{g^{2}} + \frac{h^{2}}{g^{2}}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{y^{2}}{g^{2}} + \frac{- 2 h y}{g^{2}} + \frac{h^{2}}{g^{2}}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = \frac{y^{2}}{g^{2}} - \frac{2 h y}{g^{2}} + \frac{h^{2}}{g^{2}}$