Solve for x 9+ square root of x^2-5x=12

$9 + \sqrt{x^{2} - 5 x} = 12$

Step 1

Move all terms not containing $\sqrt{x^{2} - 5 x}$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$\sqrt{x^{2} - 5 x} = 12 - 9$

Subtract $9$ from $12$ .

$\sqrt{x^{2} - 5 x} = 3$

Step 2

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

${\sqrt{x^{2} - 5 x}}^{2} = 3^{2}$

Step 3

Simplify each side of the equation.

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

${({(x^{2} - 5 x)}^{\frac{1}{2}})}^{2} = 3^{2}$

Simplify the left side.

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

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

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

${(x^{2} - 5 x)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} - 5 x)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Rewrite the expression.

${(x^{2} - 5 x)}^{1} = 3^{2}$

Simplify.

$x^{2} - 5 x = 3^{2}$

Simplify the right side.

Raise $3$ to the power of $2$ .

$x^{2} - 5 x = 9$

Step 4

Solve for $x$ .

Subtract $9$ from both sides of the equation.

$x^{2} - 5 x - 9 = 0$

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 $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 9$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot 1}$

Multiply $- 4$ by $- 9$ .

$x = \frac{5 \pm \sqrt{25 + 36}}{2 \cdot 1}$

Add $25$ and $36$ .

$x = \frac{5 \pm \sqrt{61}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{61}}{2}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{61}}{2}, \frac{5 - \sqrt{61}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{61}}{2}, \frac{5 - \sqrt{61}}{2}$

Decimal Form:

$x = 6.40512483 \dots, - 1.40512483 \dots$

Solve for x 9+ square root of x^2-5x=12

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