Solve for y square root of 2y^2-5y- square root of 21-6y=0

$\sqrt{2 y^{2} - 5 y} - \sqrt{21 - 6 y} = 0$

Step 1

Solve for $\sqrt{2 y^{2} - 5 y}$ .

Factor $3$ out of $21 - 6 y$ .

Factor $3$ out of $21$ .

$\sqrt{2 y^{2} - 5 y} - \sqrt{3 (7) - 6 y} = 0$

Factor $3$ out of $- 6 y$ .

$\sqrt{2 y^{2} - 5 y} - \sqrt{3 (7) + 3 (- 2 y)} = 0$

Factor $3$ out of $3 (7) + 3 (- 2 y)$ .

$\sqrt{2 y^{2} - 5 y} - \sqrt{3 (7 - 2 y)} = 0$

Add $\sqrt{3 (7 - 2 y)}$ to both sides of the equation.

$\sqrt{2 y^{2} - 5 y} = \sqrt{3 (7 - 2 y)}$

Step 2

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

${\sqrt{2 y^{2} - 5 y}}^{2} = {\sqrt{3 (7 - 2 y)}}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 y^{2} - 5 y}$ as ${(2 y^{2} - 5 y)}^{\frac{1}{2}}$ .

${({(2 y^{2} - 5 y)}^{\frac{1}{2}})}^{2} = {\sqrt{3 (7 - 2 y)}}^{2}$

Simplify the left side.

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

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

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

${(2 y^{2} - 5 y)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 (7 - 2 y)}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 y^{2} - 5 y)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 (7 - 2 y)}}^{2}$

Rewrite the expression.

${(2 y^{2} - 5 y)}^{1} = {\sqrt{3 (7 - 2 y)}}^{2}$

Simplify.

$2 y^{2} - 5 y = {\sqrt{3 (7 - 2 y)}}^{2}$

Simplify the right side.

Simplify ${\sqrt{3 (7 - 2 y)}}^{2}$ .

Rewrite ${\sqrt{3 (7 - 2 y)}}^{2}$ as $3 (7 - 2 y)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (7 - 2 y)}$ as ${(3 (7 - 2 y))}^{\frac{1}{2}}$ .

$2 y^{2} - 5 y = {({(3 (7 - 2 y))}^{\frac{1}{2}})}^{2}$

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

$2 y^{2} - 5 y = {(3 (7 - 2 y))}^{\frac{1}{2} \cdot 2}$

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

$2 y^{2} - 5 y = {(3 (7 - 2 y))}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 y^{2} - 5 y = {(3 (7 - 2 y))}^{\frac{2}{2}}$

Rewrite the expression.

$2 y^{2} - 5 y = {(3 (7 - 2 y))}^{1}$

Simplify.

$2 y^{2} - 5 y = 3 (7 - 2 y)$

Apply the distributive property.

$2 y^{2} - 5 y = 3 \cdot 7 + 3 (- 2 y)$

Multiply.

Multiply $3$ by $7$ .

$2 y^{2} - 5 y = 21 + 3 (- 2 y)$

Multiply $- 2$ by $3$ .

$2 y^{2} - 5 y = 21 - 6 y$

Step 4

Solve for $y$ .

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

Add $6 y$ to both sides of the equation.

$2 y^{2} - 5 y + 6 y = 21$

Add $- 5 y$ and $6 y$ .

$2 y^{2} + y = 21$

Subtract $21$ from both sides of the equation.

$2 y^{2} + y - 21 = 0$

Factor by grouping.

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 $a \cdot c = 2 \cdot - 21 = - 42$ and whose sum is $b = 1$ .

Multiply by $1$ .

$2 y^{2} + 1 y - 21 = 0$

Rewrite $1$ as $- 6$ plus $7$

$2 y^{2} + (- 6 + 7) y - 21 = 0$

Apply the distributive property.

$2 y^{2} - 6 y + 7 y - 21 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 y^{2} - 6 y) + 7 y - 21 = 0$

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

$2 y (y - 3) + 7 (y - 3) = 0$

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

$(y - 3) (2 y + 7) = 0$

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

$y - 3 = 0$

$2 y + 7 = 0$

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

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

$y - 3 = 0$

Add $3$ to both sides of the equation.

$y = 3$

Set $2 y + 7$ equal to $0$ and solve for $y$ .

Set $2 y + 7$ equal to $0$ .

$2 y + 7 = 0$

Solve $2 y + 7 = 0$ for $y$ .

Subtract $7$ from both sides of the equation.

$2 y = - 7$

Divide each term in $2 y = - 7$ by $2$ and simplify.

Divide each term in $2 y = - 7$ by $2$ .

$\frac{2 y}{2} = \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 7}{2}$

Divide $y$ by $1$ .

$y = \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{7}{2}$

The final solution is all the values that make $(y - 3) (2 y + 7) = 0$ true.

$y = 3, - \frac{7}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = 3, - \frac{7}{2}$

Decimal Form:

$y = 3, - 3.5$

Mixed Number Form:

$y = 3, - 3 \frac{1}{2}$

Solve for y square root of 2y^2-5y- square root of 21-6y=0

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