Solve Using the Quadratic Formula x(7x-4)=11

$x (7 x - 4) = 11$

Step 1

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

Simplify the left side.

Simplify $x (7 x - 4)$ .

Simplify by multiplying through.

Apply the distributive property.

$x (7 x) + x \cdot - 4 = 11$

Reorder.

Rewrite using the commutative property of multiplication.

$7 x \cdot x + x \cdot - 4 = 11$

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

$7 x \cdot x - 4 \cdot x = 11$

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

Move $x$ .

$7 (x \cdot x) - 4 \cdot x = 11$

Multiply $x$ by $x$ .

$7 x^{2} - 4 \cdot x = 11$

$7 x^{2} - 4 x = 11$

Subtract $11$ from both sides of the equation.

$7 x^{2} - 4 x - 11 = 0$

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 $a = 7$ , $b = - 4$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (7 \cdot - 11)}}{2 \cdot 7}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 7 \cdot - 11}}{2 \cdot 7}$

Multiply $- 4 \cdot 7 \cdot - 11$ .

Multiply $- 4$ by $7$ .

$x = \frac{4 \pm \sqrt{16 - 28 \cdot - 11}}{2 \cdot 7}$

Multiply $- 28$ by $- 11$ .

$x = \frac{4 \pm \sqrt{16 + 308}}{2 \cdot 7}$

Add $16$ and $308$ .

$x = \frac{4 \pm \sqrt{324}}{2 \cdot 7}$

Rewrite $324$ as $18^{2}$ .

$x = \frac{4 \pm \sqrt{18^{2}}}{2 \cdot 7}$

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

$x = \frac{4 \pm 18}{2 \cdot 7}$

Multiply $2$ by $7$ .

$x = \frac{4 \pm 18}{14}$

Simplify $\frac{4 \pm 18}{14}$ .

$x = \frac{2 \pm 9}{7}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{11}{7}, - 1$

Solve Using the Quadratic Formula x(7x-4)=11

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