Solve Using the Quadratic Formula 4x(x+1)=7

$4 x (x + 1) = 7$

Step 1

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

Simplify the left side.

Simplify $4 x (x + 1)$ .

Apply the distributive property.

$4 x \cdot x + 4 x \cdot 1 = 7$

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

Move $x$ .

$4 (x \cdot x) + 4 x \cdot 1 = 7$

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot 1 = 7$

Multiply $4$ by $1$ .

$4 x^{2} + 4 x = 7$

Subtract $7$ from both sides of the equation.

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

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

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

Step 4

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $4$ .

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

Multiply $- 16$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 112}}{2 \cdot 4}$

Add $16$ and $112$ .

$x = \frac{- 4 \pm \sqrt{128}}{2 \cdot 4}$

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$x = \frac{- 4 \pm \sqrt{64 (2)}}{2 \cdot 4}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 4 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 8 \sqrt{2}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 4 \pm 8 \sqrt{2}}{8}$

Simplify $\frac{- 4 \pm 8 \sqrt{2}}{8}$ .

$x = \frac{- 1 \pm 2 \sqrt{2}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 1 \pm 2 \sqrt{2}}{2}$

Decimal Form:

$x = 0.91421356 \dots, - 1.91421356 \dots$

Solve Using the Quadratic Formula 4x(x+1)=7

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