Solve for x square root of 5x+11-1=x

$\sqrt{5 x + 11 - 1} = x$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{5 x + 11 - 1}}^{2} = x^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 11 - 1}$ as ${(5 x + 11 - 1)}^{\frac{1}{2}}$ .

${({(5 x + 11 - 1)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(5 x + 11 - 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 x + 11 - 1)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(5 x + 11 - 1)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 x + 11 - 1)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(5 x + 11 - 1)}^{1} = x^{2}$

Subtract $1$ from $11$ .

${(5 x + 10)}^{1} = x^{2}$

Simplify.

$5 x + 10 = x^{2}$

Step 3

Solve for $x$ .

Subtract $x^{2}$ from both sides of the equation.

$5 x + 10 - x^{2} = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 1$ , $b = 5$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (- 1 \cdot 10)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 1 \cdot 10}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot 10$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 + 4 \cdot 10}}{2 \cdot - 1}$

Multiply $4$ by $10$ .

$x = \frac{- 5 \pm \sqrt{25 + 40}}{2 \cdot - 1}$

Add $25$ and $40$ .

$x = \frac{- 5 \pm \sqrt{65}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{65}}{- 2}$

Simplify $\frac{- 5 \pm \sqrt{65}}{- 2}$ .

$x = \frac{5 \pm \sqrt{65}}{2}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{65}}{2}, \frac{5 - \sqrt{65}}{2}$

Step 4

Exclude the solutions that do not make $\sqrt{5 x + 11 - 1} = x$ true.

$x = \frac{5 + \sqrt{65}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{65}}{2}$

Decimal Form:

$x = 6.53112887 \dots$

Solve for x square root of 5x+11-1=x

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