Find All Complex Solutions square root of 2x+7=x+3

$\sqrt{2 x} + 7 = x + 3$

Step 1

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

Subtract $7$ from both sides of the equation.

$\sqrt{2 x} = x + 3 - 7$

Subtract $7$ from $3$ .

$\sqrt{2 x} = x - 4$

Step 2

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

${\sqrt{2 x}}^{2} = {(x - 4)}^{2}$

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(2 x)}^{1} = {(x - 4)}^{2}$

Simplify.

$2 x = {(x - 4)}^{2}$

Simplify the right side.

Simplify ${(x - 4)}^{2}$ .

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$2 x = (x - 4) (x - 4)$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$2 x = x (x - 4) - 4 (x - 4)$

Apply the distributive property.

$2 x = x \cdot x + x \cdot - 4 - 4 (x - 4)$

Apply the distributive property.

$2 x = x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 4$ to the left of $x$ .

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

Multiply $- 4$ by $- 4$ .

$2 x = x^{2} - 4 x - 4 x + 16$

Subtract $4 x$ from $- 4 x$ .

$2 x = x^{2} - 8 x + 16$

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.

$x^{2} - 8 x + 16 = 2 x$

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

Subtract $2 x$ from both sides of the equation.

$x^{2} - 8 x + 16 - 2 x = 0$

Subtract $2 x$ from $- 8 x$ .

$x^{2} - 10 x + 16 = 0$

Factor $x^{2} - 10 x + 16$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $16$ and whose sum is $- 10$ .

$- 8, - 2$

Write the factored form using these integers.

$(x - 8) (x - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 8 = 0$

$x - 2 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 8) (x - 2) = 0$ true.

$x = 8, 2$

Step 5

Exclude the solutions that do not make $\sqrt{2 x} + 7 = x + 3$ true.

$x = 8$

Find All Complex Solutions square root of 2x+7=x+3

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