Solve by Factoring 35k^2-22k+7=4

$35 k^{2} - 22 k + 7 = 4$

Step 1

Subtract $4$ from both sides of the equation.

$35 k^{2} - 22 k + 7 - 4 = 0$

Step 2

Subtract $4$ from $7$ .

$35 k^{2} - 22 k + 3 = 0$

Step 3

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 $a \cdot c = 35 \cdot 3 = 105$ and whose sum is $b = - 22$ .

Factor $- 22$ out of $- 22 k$ .

$35 k^{2} - 22 k + 3 = 0$

Rewrite $- 22$ as $- 7$ plus $- 15$

$35 k^{2} + (- 7 - 15) k + 3 = 0$

Apply the distributive property.

$35 k^{2} - 7 k - 15 k + 3 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(35 k^{2} - 7 k) - 15 k + 3 = 0$

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

$7 k (5 k - 1) - 3 (5 k - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 k - 1$ .

$(5 k - 1) (7 k - 3) = 0$

Step 4

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

$5 k - 1 = 0$

$7 k - 3 = 0$

Step 5

Set $5 k - 1$ equal to $0$ and solve for $k$ .

Set $5 k - 1$ equal to $0$ .

$5 k - 1 = 0$

Solve $5 k - 1 = 0$ for $k$ .

Add $1$ to both sides of the equation.

$5 k = 1$

Divide each term in $5 k = 1$ by $5$ and simplify.

Divide each term in $5 k = 1$ by $5$ .

$\frac{5 k}{5} = \frac{1}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 k}{5} = \frac{1}{5}$

Divide $k$ by $1$ .

$k = \frac{1}{5}$

Step 6

Set $7 k - 3$ equal to $0$ and solve for $k$ .

Set $7 k - 3$ equal to $0$ .

$7 k - 3 = 0$

Solve $7 k - 3 = 0$ for $k$ .

Add $3$ to both sides of the equation.

$7 k = 3$

Divide each term in $7 k = 3$ by $7$ and simplify.

Divide each term in $7 k = 3$ by $7$ .

$\frac{7 k}{7} = \frac{3}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 k}{7} = \frac{3}{7}$

Divide $k$ by $1$ .

$k = \frac{3}{7}$

Step 7

The final solution is all the values that make $(5 k - 1) (7 k - 3) = 0$ true.

$k = \frac{1}{5}, \frac{3}{7}$

Solve by Factoring 35k^2-22k+7=4

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