Solve by Completing the Square 49x^2+28x=22

$49 x^{2} + 28 x = 22$

Step 1

Divide each term in $49 x^{2} + 28 x = 22$ by $49$ and simplify.

Divide each term in $49 x^{2} + 28 x = 22$ by $49$ .

$\frac{49 x^{2}}{49} + \frac{28 x}{49} = \frac{22}{49}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $49$ .

Cancel the common factor.

$\frac{49 x^{2}}{49} + \frac{28 x}{49} = \frac{22}{49}$

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

$x^{2} + \frac{28 x}{49} = \frac{22}{49}$

Cancel the common factor of $28$ and $49$ .

Factor $7$ out of $28 x$ .

$x^{2} + \frac{7 (4 x)}{49} = \frac{22}{49}$

Cancel the common factors.

Factor $7$ out of $49$ .

$x^{2} + \frac{7 (4 x)}{7 (7)} = \frac{22}{49}$

Cancel the common factor.

$x^{2} + \frac{7 (4 x)}{7 \cdot 7} = \frac{22}{49}$

Rewrite the expression.

$x^{2} + \frac{4 x}{7} = \frac{22}{49}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{2}{7})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + \frac{4 x}{7} + {(\frac{2}{7})}^{2} = \frac{22}{49} + {(\frac{2}{7})}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{2}{7}$ .

$x^{2} + \frac{4 x}{7} + \frac{2^{2}}{7^{2}} = \frac{22}{49} + {(\frac{2}{7})}^{2}$

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

$x^{2} + \frac{4 x}{7} + \frac{4}{7^{2}} = \frac{22}{49} + {(\frac{2}{7})}^{2}$

Raise $7$ to the power of $2$ .

$x^{2} + \frac{4 x}{7} + \frac{4}{49} = \frac{22}{49} + {(\frac{2}{7})}^{2}$

Simplify the right side.

Simplify $\frac{22}{49} + {(\frac{2}{7})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{2}{7}$ .

$x^{2} + \frac{4 x}{7} + \frac{4}{49} = \frac{22}{49} + \frac{2^{2}}{7^{2}}$

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

$x^{2} + \frac{4 x}{7} + \frac{4}{49} = \frac{22}{49} + \frac{4}{7^{2}}$

Raise $7$ to the power of $2$ .

$x^{2} + \frac{4 x}{7} + \frac{4}{49} = \frac{22}{49} + \frac{4}{49}$

Combine fractions.

Combine the numerators over the common denominator.

$x^{2} + \frac{4 x}{7} + \frac{4}{49} = \frac{22 + 4}{49}$

Add $22$ and $4$ .

$x^{2} + \frac{4 x}{7} + \frac{4}{49} = \frac{26}{49}$

Step 5

Factor the perfect trinomial square into ${(x + \frac{2}{7})}^{2}$ .

${(x + \frac{2}{7})}^{2} = \frac{26}{49}$

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.

$x + \frac{2}{7} = \pm \sqrt{\frac{26}{49}}$

Simplify $\pm \sqrt{\frac{26}{49}}$ .

Rewrite $\sqrt{\frac{26}{49}}$ as $\frac{\sqrt{26}}{\sqrt{49}}$ .

$x + \frac{2}{7} = \pm \frac{\sqrt{26}}{\sqrt{49}}$

Simplify the denominator.

Rewrite $49$ as $7^{2}$ .

$x + \frac{2}{7} = \pm \frac{\sqrt{26}}{\sqrt{7^{2}}}$

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

$x + \frac{2}{7} = \pm \frac{\sqrt{26}}{7}$

Subtract $\frac{2}{7}$ from both sides of the equation.

$x = \pm \frac{\sqrt{26}}{7} - \frac{2}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{26}}{7} - \frac{2}{7}$

Decimal Form:

$x = 0.44271707 \dots, - 1.01414564 \dots$

Solve by Completing the Square 49x^2+28x=22

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