Solve the Rational Equation for x square root of x-2x+7=0

$\sqrt{x} - 2 x + 7 = 0$

Step 1

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

Add $2 x$ to both sides of the equation.

$\sqrt{x} + 7 = 2 x$

Subtract $7$ from both sides of the equation.

$\sqrt{x} = 2 x - 7$

Step 2

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

${\sqrt{x}}^{2} = {(2 x - 7)}^{2}$

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x^{1} = {(2 x - 7)}^{2}$

Simplify.

$x = {(2 x - 7)}^{2}$

Simplify the right side.

Simplify ${(2 x - 7)}^{2}$ .

Rewrite ${(2 x - 7)}^{2}$ as $(2 x - 7) (2 x - 7)$ .

$x = (2 x - 7) (2 x - 7)$

Expand $(2 x - 7) (2 x - 7)$ using the FOIL Method.

Apply the distributive property.

$x = 2 x (2 x - 7) - 7 (2 x - 7)$

Apply the distributive property.

$x = 2 x (2 x) + 2 x \cdot - 7 - 7 (2 x - 7)$

Apply the distributive property.

$x = 2 x (2 x) + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x = 2 \cdot 2 x \cdot x + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

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

Move $x$ .

$x = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

Multiply $x$ by $x$ .

$x = 2 \cdot 2 x^{2} + 2 x \cdot - 7 - 7 (2 x) - 7 \cdot - 7$

Multiply $2$ by $2$ .

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

Multiply $- 7$ by $2$ .

$x = 4 x^{2} - 14 x - 7 (2 x) - 7 \cdot - 7$

Multiply $2$ by $- 7$ .

$x = 4 x^{2} - 14 x - 14 x - 7 \cdot - 7$

Multiply $- 7$ by $- 7$ .

$x = 4 x^{2} - 14 x - 14 x + 49$

Subtract $14 x$ from $- 14 x$ .

$x = 4 x^{2} - 28 x + 49$

Step 4

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 x^{2} - 28 x + 49 = x$

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$4 x^{2} - 28 x + 49 - x = 0$

Subtract $x$ from $- 28 x$ .

$4 x^{2} - 29 x + 49 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 4$ , $b = - 29$ , and $c = 49$ into the quadratic formula and solve for $x$ .

$\frac{29 \pm \sqrt{{(- 29)}^{2} - 4 \cdot (4 \cdot 49)}}{2 \cdot 4}$

Simplify.

Simplify the numerator.

Raise $- 29$ to the power of $2$ .

$x = \frac{29 \pm \sqrt{841 - 4 \cdot 4 \cdot 49}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot 49$ .

Multiply $- 4$ by $4$ .

$x = \frac{29 \pm \sqrt{841 - 16 \cdot 49}}{2 \cdot 4}$

Multiply $- 16$ by $49$ .

$x = \frac{29 \pm \sqrt{841 - 784}}{2 \cdot 4}$

Subtract $784$ from $841$ .

$x = \frac{29 \pm \sqrt{57}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{29 \pm \sqrt{57}}{8}$

The final answer is the combination of both solutions.

$x = \frac{29 + \sqrt{57}}{8}, \frac{29 - \sqrt{57}}{8}$

Step 5

Exclude the solutions that do not make $\sqrt{x} - 2 x + 7 = 0$ true.

$x = \frac{29 + \sqrt{57}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{29 + \sqrt{57}}{8}$

Decimal Form:

$x = 4.56872930 \dots$

Solve the Rational Equation for x square root of x-2x+7=0

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