Solve by Factoring 24x^2-x=10x-1

$24 x^{2} - x = 10 x - 1$

Step 1

Move all the expressions to the left side of the equation.

Subtract $10 x$ from both sides of the equation.

$24 x^{2} - x - 10 x = - 1$

Add $1$ to both sides of the equation.

$24 x^{2} - x - 10 x + 1 = 0$

Step 2

Subtract $10 x$ from $- x$ .

$24 x^{2} - 11 x + 1 = 0$

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 $a \cdot c = 24 \cdot 1 = 24$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$24 x^{2} - 11 x + 1 = 0$

Rewrite $- 11$ as $- 3$ plus $- 8$

$24 x^{2} + (- 3 - 8) x + 1 = 0$

Apply the distributive property.

$24 x^{2} - 3 x - 8 x + 1 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(24 x^{2} - 3 x) - 8 x + 1 = 0$

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

$3 x (8 x - 1) - (8 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $8 x - 1$ .

$(8 x - 1) (3 x - 1) = 0$

Step 4

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

$8 x - 1 = 0$

$3 x - 1 = 0$

Step 5

Set $8 x - 1$ equal to $0$ and solve for $x$ .

Set $8 x - 1$ equal to $0$ .

$8 x - 1 = 0$

Solve $8 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$8 x = 1$

Divide each term in $8 x = 1$ by $8$ and simplify.

Divide each term in $8 x = 1$ by $8$ .

$\frac{8 x}{8} = \frac{1}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{1}{8}$

Divide $x$ by $1$ .

$x = \frac{1}{8}$

Step 6

Set $3 x - 1$ equal to $0$ and solve for $x$ .

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Solve $3 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$3 x = 1$

Divide each term in $3 x = 1$ by $3$ and simplify.

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 7

The final solution is all the values that make $(8 x - 1) (3 x - 1) = 0$ true.

$x = \frac{1}{8}, \frac{1}{3}$

Solve by Factoring 24x^2-x=10x-1

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