Solve by Completing the Square 4x^2+8x=7

$4 x^{2} + 8 x = 7$

Step 1

Divide each term in $4 x^{2} + 8 x = 7$ by $4$ and simplify.

Divide each term in $4 x^{2} + 8 x = 7$ by $4$ .

$\frac{4 x^{2}}{4} + \frac{8 x}{4} = \frac{7}{4}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} + \frac{8 x}{4} = \frac{7}{4}$

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{8 x}{4} = \frac{7}{4}$

Cancel the common factor of $8$ and $4$ .

Factor $4$ out of $8 x$ .

$x^{2} + \frac{4 (2 x)}{4} = \frac{7}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$x^{2} + \frac{4 (2 x)}{4 (1)} = \frac{7}{4}$

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 x$ by $1$ .

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

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(1)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 2 x + {(1)}^{2} = \frac{7}{4} + {(1)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

One to any power is one.

$x^{2} + 2 x + 1 = \frac{7}{4} + {(1)}^{2}$

Simplify the right side.

Simplify $\frac{7}{4} + {(1)}^{2}$ .

One to any power is one.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $7$ and $4$ .

$x^{2} + 2 x + 1 = \frac{11}{4}$

Step 5

Factor the perfect trinomial square into ${(x + 1)}^{2}$ .

${(x + 1)}^{2} = \frac{11}{4}$

Step 6

Solve the equation for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 1 = \pm \sqrt{\frac{11}{4}}$

Simplify $\pm \sqrt{\frac{11}{4}}$ .

Rewrite $\sqrt{\frac{11}{4}}$ as $\frac{\sqrt{11}}{\sqrt{4}}$ .

$x + 1 = \pm \frac{\sqrt{11}}{\sqrt{4}}$

Simplify the denominator.

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

$x + 1 = \pm \frac{\sqrt{11}}{\sqrt{2^{2}}}$

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

$x + 1 = \pm \frac{\sqrt{11}}{2}$

Subtract $1$ from both sides of the equation.

$x = \pm \frac{\sqrt{11}}{2} - 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{11}}{2} - 1$

Decimal Form:

$x = 0.65831239 \dots, - 2.65831239 \dots$

Solve by Completing the Square 4x^2+8x=7

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