Solve Using the Quadratic Formula (x-1)(10x-4)=-1

$(x - 1) (10 x - 4) = - 1$

Step 1

Move all terms to the left side of the equation and simplify.

Simplify the left side.

Simplify $(x - 1) (10 x - 4)$ .

Expand $(x - 1) (10 x - 4)$ using the FOIL Method.

Apply the distributive property.

$x (10 x - 4) - 1 (10 x - 4) = - 1$

Apply the distributive property.

$x (10 x) + x \cdot - 4 - 1 (10 x - 4) = - 1$

Apply the distributive property.

$x (10 x) + x \cdot - 4 - 1 (10 x) - 1 \cdot - 4 = - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$10 x \cdot x + x \cdot - 4 - 1 (10 x) - 1 \cdot - 4 = - 1$

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

Move $x$ .

$10 (x \cdot x) + x \cdot - 4 - 1 (10 x) - 1 \cdot - 4 = - 1$

Multiply $x$ by $x$ .

$10 x^{2} + x \cdot - 4 - 1 (10 x) - 1 \cdot - 4 = - 1$

Move $- 4$ to the left of $x$ .

$10 x^{2} - 4 \cdot x - 1 (10 x) - 1 \cdot - 4 = - 1$

Multiply $10$ by $- 1$ .

$10 x^{2} - 4 x - 10 x - 1 \cdot - 4 = - 1$

Multiply $- 1$ by $- 4$ .

$10 x^{2} - 4 x - 10 x + 4 = - 1$

Subtract $10 x$ from $- 4 x$ .

$10 x^{2} - 14 x + 4 = - 1$

Add $1$ to both sides of the equation.

$10 x^{2} - 14 x + 4 + 1 = 0$

Add $4$ and $1$ .

$10 x^{2} - 14 x + 5 = 0$

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 $a = 10$ , $b = - 14$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (10 \cdot 5)}}{2 \cdot 10}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 10 \cdot 5}}{2 \cdot 10}$

Multiply $- 4 \cdot 10 \cdot 5$ .

Multiply $- 4$ by $10$ .

$x = \frac{14 \pm \sqrt{196 - 40 \cdot 5}}{2 \cdot 10}$

Multiply $- 40$ by $5$ .

$x = \frac{14 \pm \sqrt{196 - 200}}{2 \cdot 10}$

Subtract $200$ from $196$ .

$x = \frac{14 \pm \sqrt{- 4}}{2 \cdot 10}$

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{14 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 10}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{14 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 10}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{14 \pm i \cdot \sqrt{4}}{2 \cdot 10}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{14 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 10}$

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

$x = \frac{14 \pm i \cdot 2}{2 \cdot 10}$

Move $2$ to the left of $i$ .

$x = \frac{14 \pm 2 i}{2 \cdot 10}$

Multiply $2$ by $10$ .

$x = \frac{14 \pm 2 i}{20}$

Simplify $\frac{14 \pm 2 i}{20}$ .

$x = \frac{7 \pm i}{10}$

Solve Using the Quadratic Formula (x-1)(10x-4)=-1

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