Solve Using the Quadratic Formula 7x(x-2)=5

$7 x (x - 2) = 5$

Step 1

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

Simplify the left side.

Simplify $7 x (x - 2)$ .

Apply the distributive property.

$7 x \cdot x + 7 x \cdot - 2 = 5$

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

Move $x$ .

$7 (x \cdot x) + 7 x \cdot - 2 = 5$

Multiply $x$ by $x$ .

$7 x^{2} + 7 x \cdot - 2 = 5$

Multiply $- 2$ by $7$ .

$7 x^{2} - 14 x = 5$

Subtract $5$ from both sides of the equation.

$7 x^{2} - 14 x - 5 = 0$

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 $a = 7$ , $b = - 14$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (7 \cdot - 5)}}{2 \cdot 7}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 7 \cdot - 5}}{2 \cdot 7}$

Multiply $- 4 \cdot 7 \cdot - 5$ .

Multiply $- 4$ by $7$ .

$x = \frac{14 \pm \sqrt{196 - 28 \cdot - 5}}{2 \cdot 7}$

Multiply $- 28$ by $- 5$ .

$x = \frac{14 \pm \sqrt{196 + 140}}{2 \cdot 7}$

Add $196$ and $140$ .

$x = \frac{14 \pm \sqrt{336}}{2 \cdot 7}$

Rewrite $336$ as $4^{2} \cdot 21$ .

Factor $16$ out of $336$ .

$x = \frac{14 \pm \sqrt{16 (21)}}{2 \cdot 7}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{14 \pm \sqrt{4^{2} \cdot 21}}{2 \cdot 7}$

Pull terms out from under the radical.

$x = \frac{14 \pm 4 \sqrt{21}}{2 \cdot 7}$

Multiply $2$ by $7$ .

$x = \frac{14 \pm 4 \sqrt{21}}{14}$

Simplify $\frac{14 \pm 4 \sqrt{21}}{14}$ .

$x = \frac{7 \pm 2 \sqrt{21}}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 \pm 2 \sqrt{21}}{7}$

Decimal Form:

$x = 2.30930734 \dots, - 0.30930734 \dots$

Solve Using the Quadratic Formula 7x(x-2)=5

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