Solve by Factoring 17y+5y^2=40

$17 y + 5 y^{2} = 40$

Step 1

Subtract $40$ from both sides of the equation.

$17 y + 5 y^{2} - 40 = 0$

Step 2

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$17 u + 5 u^{2} - 40 = 0$

Step 3

Factor by grouping.

Reorder terms.

$5 u^{2} + 17 u - 40 = 0$

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 $a \cdot c = 5 \cdot - 40 = - 200$ and whose sum is $b = 17$ .

Factor $17$ out of $17 u$ .

$5 u^{2} + 17 (u) - 40 = 0$

Rewrite $17$ as $- 8$ plus $25$

$5 u^{2} + (- 8 + 25) u - 40 = 0$

Apply the distributive property.

$5 u^{2} - 8 u + 25 u - 40 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 u^{2} - 8 u) + 25 u - 40 = 0$

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

$u (5 u - 8) + 5 (5 u - 8) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 u - 8$ .

$(5 u - 8) (u + 5) = 0$

Step 4

Replace all occurrences of $u$ with $y$ .

$(5 y - 8) (y + 5) = 0$

Step 5

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

$5 y - 8 = 0$

$y + 5 = 0$

Step 6

Set $5 y - 8$ equal to $0$ and solve for $y$ .

Set $5 y - 8$ equal to $0$ .

$5 y - 8 = 0$

Solve $5 y - 8 = 0$ for $y$ .

Add $8$ to both sides of the equation.

$5 y = 8$

Divide each term in $5 y = 8$ by $5$ and simplify.

Divide each term in $5 y = 8$ by $5$ .

$\frac{5 y}{5} = \frac{8}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{8}{5}$

Divide $y$ by $1$ .

$y = \frac{8}{5}$

Step 7

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

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

$y + 5 = 0$

Subtract $5$ from both sides of the equation.

$y = - 5$

Step 8

The final solution is all the values that make $(5 y - 8) (y + 5) = 0$ true.

$y = \frac{8}{5}, - 5$

Solve by Factoring 17y+5y^2=40

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