Basic Math Examples

$\sqrt{15 - 4 y} - 2 y = 0$

Step 1

Add $2 y$ to both sides of the equation.

$\sqrt{15 - 4 y} = 2 y$

Step 2

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

${\sqrt{15 - 4 y}}^{2} = {(2 y)}^{2}$

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 $\sqrt{15 - 4 y}$ as ${(15 - 4 y)}^{\frac{1}{2}}$ .

${({(15 - 4 y)}^{\frac{1}{2}})}^{2} = {(2 y)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(15 - 4 y)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(15 - 4 y)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(15 - 4 y)}^{\frac{1}{2} \cdot 2} = {(2 y)}^{2}$

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.

${(15 - 4 y)}^{\frac{1}{2} \cdot 2} = {(2 y)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(15 - 4 y)}^{1} = {(2 y)}^{2}$

Step 3.2.1.2

Simplify.

$15 - 4 y = {(2 y)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Apply the product rule to $2 y$ .

$15 - 4 y = 2^{2} y^{2}$

Step 3.3.1.2

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

$15 - 4 y = 4 y^{2}$

Step 4

Solve for $y$ .

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Step 4.1

Subtract $4 y^{2}$ from both sides of the equation.

$15 - 4 y - 4 y^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $- 1$ out of $15 - 4 y - 4 y^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $15$ .

$- 4 y - 4 y^{2} + 15 = 0$

Step 4.2.1.1.2

Reorder $- 4 y$ and $- 4 y^{2}$ .

$- 4 y^{2} - 4 y + 15 = 0$

Step 4.2.1.2

Factor $- 1$ out of $- 4 y^{2}$ .

$- (4 y^{2}) - 4 y + 15 = 0$

Step 4.2.1.3

Factor $- 1$ out of $- 4 y$ .

$- (4 y^{2}) - (4 y) + 15 = 0$

Step 4.2.1.4

Rewrite $15$ as $- 1 (- 15)$ .

$- (4 y^{2}) - (4 y) - 1 \cdot - 15 = 0$

Step 4.2.1.5

Factor $- 1$ out of $- (4 y^{2}) - (4 y)$ .

$- (4 y^{2} + 4 y) - 1 \cdot - 15 = 0$

Step 4.2.1.6

Factor $- 1$ out of $- (4 y^{2} + 4 y) - 1 (- 15)$ .

$- (4 y^{2} + 4 y - 15) = 0$

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor by grouping.

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Step 4.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 15 = - 60$ and whose sum is $b = 4$ .

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Step 4.2.2.1.1.1

Factor $4$ out of $4 y$ .

$- (4 y^{2} + 4 (y) - 15) = 0$

Step 4.2.2.1.1.2

Rewrite $4$ as $- 6$ plus $10$

$- (4 y^{2} + (- 6 + 10) y - 15) = 0$

Step 4.2.2.1.1.3

Apply the distributive property.

$- (4 y^{2} - 6 y + 10 y - 15) = 0$

Step 4.2.2.1.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((4 y^{2} - 6 y) + 10 y - 15) = 0$

Step 4.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (2 y (2 y - 3) + 5 (2 y - 3)) = 0$

Step 4.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 y - 3$ .

$- ((2 y - 3) (2 y + 5)) = 0$

Step 4.2.2.2

Remove unnecessary parentheses.

$- (2 y - 3) (2 y + 5) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 y - 3 = 0$

$2 y + 5 = 0$

Step 4.4

Set $2 y - 3$ equal to $0$ and solve for $y$ .

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Step 4.4.1

Set $2 y - 3$ equal to $0$ .

$2 y - 3 = 0$

Step 4.4.2

Solve $2 y - 3 = 0$ for $y$ .

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Step 4.4.2.1

Add $3$ to both sides of the equation.

$2 y = 3$

Step 4.4.2.2

Divide each term in $2 y = 3$ by $2$ and simplify.

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Step 4.4.2.2.1

Divide each term in $2 y = 3$ by $2$ .

$\frac{2 y}{2} = \frac{3}{2}$

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{3}{2}$

Step 4.4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{3}{2}$

Step 4.5

Set $2 y + 5$ equal to $0$ and solve for $y$ .

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Step 4.5.1

Set $2 y + 5$ equal to $0$ .

$2 y + 5 = 0$

Step 4.5.2

Solve $2 y + 5 = 0$ for $y$ .

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Step 4.5.2.1

Subtract $5$ from both sides of the equation.

$2 y = - 5$

Step 4.5.2.2

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

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Step 4.5.2.2.1

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

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

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

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

Step 4.5.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 4.5.2.2.3

Simplify the right side.

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Step 4.5.2.2.3.1

Move the negative in front of the fraction.

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

Step 4.6

The final solution is all the values that make $- (2 y - 3) (2 y + 5) = 0$ true.

$y = \frac{3}{2}, - \frac{5}{2}$

Step 5

Exclude the solutions that do not make $\sqrt{15 - 4 y} - 2 y = 0$ true.

$y = \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{3}{2}$

Decimal Form:

$y = 1.5$

Mixed Number Form:

$y = 1 \frac{1}{2}$