Solve for x (4x-1)^2=20x-5

${(4 x - 1)}^{2} = 20 x - 5$

Step 1

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

Subtract $20 x$ from both sides of the equation.

${(4 x - 1)}^{2} - 20 x = - 5$

Simplify each term.

Rewrite ${(4 x - 1)}^{2}$ as $(4 x - 1) (4 x - 1)$ .

$(4 x - 1) (4 x - 1) - 20 x = - 5$

Expand $(4 x - 1) (4 x - 1)$ using the FOIL Method.

Apply the distributive property.

$4 x (4 x - 1) - 1 (4 x - 1) - 20 x = - 5$

Apply the distributive property.

$4 x (4 x) + 4 x \cdot - 1 - 1 (4 x - 1) - 20 x = - 5$

Apply the distributive property.

$4 x (4 x) + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1 - 20 x = - 5$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 \cdot 4 x \cdot x + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1 - 20 x = - 5$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$4 \cdot 4 (x \cdot x) + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1 - 20 x = - 5$

Multiply $x$ by $x$ .

$4 \cdot 4 x^{2} + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1 - 20 x = - 5$

Multiply $4$ by $4$ .

$16 x^{2} + 4 x \cdot - 1 - 1 (4 x) - 1 \cdot - 1 - 20 x = - 5$

Multiply $- 1$ by $4$ .

$16 x^{2} - 4 x - 1 (4 x) - 1 \cdot - 1 - 20 x = - 5$

Multiply $4$ by $- 1$ .

$16 x^{2} - 4 x - 4 x - 1 \cdot - 1 - 20 x = - 5$

Multiply $- 1$ by $- 1$ .

$16 x^{2} - 4 x - 4 x + 1 - 20 x = - 5$

Subtract $4 x$ from $- 4 x$ .

$16 x^{2} - 8 x + 1 - 20 x = - 5$

Subtract $20 x$ from $- 8 x$ .

$16 x^{2} - 28 x + 1 = - 5$

Step 2

Add $5$ to both sides of the equation.

$16 x^{2} - 28 x + 1 + 5 = 0$

Step 3

Add $1$ and $5$ .

$16 x^{2} - 28 x + 6 = 0$

Step 4

Factor the left side of the equation.

Factor $2$ out of $16 x^{2} - 28 x + 6$ .

Factor $2$ out of $16 x^{2}$ .

$2 (8 x^{2}) - 28 x + 6 = 0$

Factor $2$ out of $- 28 x$ .

$2 (8 x^{2}) + 2 (- 14 x) + 6 = 0$

Factor $2$ out of $6$ .

$2 (8 x^{2}) + 2 (- 14 x) + 2 (3) = 0$

Factor $2$ out of $2 (8 x^{2}) + 2 (- 14 x)$ .

$2 (8 x^{2} - 14 x) + 2 (3) = 0$

Factor $2$ out of $2 (8 x^{2} - 14 x) + 2 (3)$ .

$2 (8 x^{2} - 14 x + 3) = 0$

Factor.

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 $a \cdot c = 8 \cdot 3 = 24$ and whose sum is $b = - 14$ .

Factor $- 14$ out of $- 14 x$ .

$2 (8 x^{2} - 14 x + 3) = 0$

Rewrite $- 14$ as $- 2$ plus $- 12$

$2 (8 x^{2} + (- 2 - 12) x + 3) = 0$

Apply the distributive property.

$2 (8 x^{2} - 2 x - 12 x + 3) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((8 x^{2} - 2 x) - 12 x + 3) = 0$

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

$2 (2 x (4 x - 1) - 3 (4 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$2 ((4 x - 1) (2 x - 3)) = 0$

Remove unnecessary parentheses.

$2 (4 x - 1) (2 x - 3) = 0$

Step 5

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

$4 x - 1 = 0$

$2 x - 3 = 0$

Step 6

Set $4 x - 1$ equal to $0$ and solve for $x$ .

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Solve $4 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 7

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 8

The final solution is all the values that make $2 (4 x - 1) (2 x - 3) = 0$ true.

$x = \frac{1}{4}, \frac{3}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}, \frac{3}{2}$

Decimal Form:

$x = 0.25, 1.5$

Mixed Number Form:

$x = \frac{1}{4}, 1 \frac{1}{2}$

Solve for x (4x-1)^2=20x-5

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