Solve for y 1/7y^2-y+1/28=0

$\frac{1}{7} y^{2} - y + \frac{1}{28} = 0$

Step 1

Combine $\frac{1}{7}$ and $y^{2}$ .

$\frac{y^{2}}{7} - y + \frac{1}{28} = 0$

Step 2

Multiply through by the least common denominator $28$ , then simplify.

Apply the distributive property.

$28 (\frac{y^{2}}{7}) + 28 (- y) + 28 (\frac{1}{28}) = 0$

Simplify.

Cancel the common factor of $7$ .

Factor $7$ out of $28$ .

$7 (4) (\frac{y^{2}}{7}) + 28 (- y) + 28 (\frac{1}{28}) = 0$

Cancel the common factor.

$7 \cdot (4 (\frac{y^{2}}{7})) + 28 (- y) + 28 (\frac{1}{28}) = 0$

Rewrite the expression.

$4 y^{2} + 28 (- y) + 28 (\frac{1}{28}) = 0$

Multiply $- 1$ by $28$ .

$4 y^{2} - 28 y + 28 (\frac{1}{28}) = 0$

Cancel the common factor of $28$ .

Cancel the common factor.

$4 y^{2} - 28 y + 28 (\frac{1}{28}) = 0$

Rewrite the expression.

$4 y^{2} - 28 y + 1 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = - 28$ , and $c = 1$ into the quadratic formula and solve for $y$ .

$\frac{28 \pm \sqrt{{(- 28)}^{2} - 4 \cdot (4 \cdot 1)}}{2 \cdot 4}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 28$ to the power of $2$ .

$y = \frac{28 \pm \sqrt{784 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$y = \frac{28 \pm \sqrt{784 - 16 \cdot 1}}{2 \cdot 4}$

Multiply $- 16$ by $1$ .

$y = \frac{28 \pm \sqrt{784 - 16}}{2 \cdot 4}$

Subtract $16$ from $784$ .

$y = \frac{28 \pm \sqrt{768}}{2 \cdot 4}$

Rewrite $768$ as $16^{2} \cdot 3$ .

Factor $256$ out of $768$ .

$y = \frac{28 \pm \sqrt{256 (3)}}{2 \cdot 4}$

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

$y = \frac{28 \pm \sqrt{16^{2} \cdot 3}}{2 \cdot 4}$

Pull terms out from under the radical.

$y = \frac{28 \pm 16 \sqrt{3}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$y = \frac{28 \pm 16 \sqrt{3}}{8}$

Simplify $\frac{28 \pm 16 \sqrt{3}}{8}$ .

$y = \frac{7 \pm 4 \sqrt{3}}{2}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{7 + 4 \sqrt{3}}{2}, \frac{7 - 4 \sqrt{3}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = \frac{7 + 4 \sqrt{3}}{2}, \frac{7 - 4 \sqrt{3}}{2}$

Decimal Form:

$y = 6.96410161 \dots, 0.03589838 \dots$

Solve for y 1/7y^2-y+1/28=0

Related Questions