Solve Using the Quadratic Formula (2x-4)(2x-1)=0

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

Step 1

Simplify the left side.

Simplify $(2 x - 4) (2 x - 1)$ .

Expand $(2 x - 4) (2 x - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 1 - 4 (2 x - 1) = 0$

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 = 0$

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

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 = 0$

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 = 0$

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 1 - 4 (2 x) - 4 \cdot - 1 = 0$

Multiply $- 1$ by $2$ .

$4 x^{2} - 2 x - 4 (2 x) - 4 \cdot - 1 = 0$

Multiply $2$ by $- 4$ .

$4 x^{2} - 2 x - 8 x - 4 \cdot - 1 = 0$

Multiply $- 4$ by $- 1$ .

$4 x^{2} - 2 x - 8 x + 4 = 0$

Subtract $8 x$ from $- 2 x$ .

$4 x^{2} - 10 x + 4 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 4$ , $b = - 10$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (4 \cdot 4)}}{2 \cdot 4}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 4 \cdot 4}}{2 \cdot 4}$

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

Multiply $- 4$ by $4$ .

$x = \frac{10 \pm \sqrt{100 - 16 \cdot 4}}{2 \cdot 4}$

Multiply $- 16$ by $4$ .

$x = \frac{10 \pm \sqrt{100 - 64}}{2 \cdot 4}$

Subtract $64$ from $100$ .

$x = \frac{10 \pm \sqrt{36}}{2 \cdot 4}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{10 \pm \sqrt{6^{2}}}{2 \cdot 4}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{10 \pm 6}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{10 \pm 6}{8}$

Simplify $\frac{10 \pm 6}{8}$ .

$x = \frac{5 \pm 3}{4}$

Step 5

The final answer is the combination of both solutions.

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

Solve Using the Quadratic Formula (2x-4)(2x-1)=0

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