Solve the Rational Equation for x square root of 1-5x=1+ square root of 6-x

Solution:

Step 1:

Square both sides to eliminate the square root on the left side.

$(\sqrt{1-5x}{)}^2=(1+\sqrt{6-x}{)}^2$

Step 2:

Expand and simplify both sides of the equation.

Step 2.1:

Convert the square root to a power.

$(1-5x{)}^{\frac{1}{2}}=(1+\sqrt{6-x}{)}^2$

Step 2.2:

Simplify the left side by squaring the expression.

Step 2.2.1:

Square the expression inside the brackets.

$(1-5x{)}^{\frac{1}{2}\cdot 2}=(1+\sqrt{6-x}{)}^2$

Step 2.2.2:

Cancel out the exponent of 2 with the square root.

$(1-5x{)}^1=(1+\sqrt{6-x}{)}^2$

Step 2.3:

Simplify the right side by expanding the binomial.

Step 2.3.1:

Expand using the binomial theorem.

$1-5x=(1+\sqrt{6-x})\cdot (1+\sqrt{6-x})$

Step 2.3.2:

Apply the distributive property (FOIL method).

$1-5x=1\cdot 1+1\cdot \sqrt{6-x}+\sqrt{6-x}\cdot 1+\sqrt{6-x}\cdot \sqrt{6-x}$

Step 2.3.3:

Combine like terms and simplify.

$1-5x=1+2\sqrt{6-x}+(6-x)$

Step 3:

Isolate the term with the square root.

Step 3.1:

Rearrange the equation.

$2\sqrt{6-x}-x=1-5x-7$

Step 3.2:

Combine like terms.

$2\sqrt{6-x}=-4x-6$

Step 4:

Square both sides again to eliminate the square root on the left side.

$(2\sqrt{6-x}{)}^2=(-4x-6{)}^2$

Step 5:

Expand and simplify both sides of the equation.

Step 5.1:

Convert the square root to a power.

$(2(6-x{)}^{\frac{1}{2}}{)}^2=(-4x-6{)}^2$

Step 5.2:

Simplify the left side by squaring the expression.

Step 5.2.1:

Square the expression inside the brackets.

$4(6-x{)}^{\frac{1}{2}\cdot 2}=(-4x-6{)}^2$

Step 5.2.2:

Cancel out the exponent of 2 with the square root.

$4(6-x{)}^1=(-4x-6{)}^2$

Step 5.3:

Simplify the right side by expanding the binomial.

Step 5.3.1:

Expand using the binomial theorem.

$24-4x=(-4x-6)\cdot (-4x-6)$

Step 6:

Solve the resulting quadratic equation for x.

Step 6.1:

Move all terms to one side.

$16x^2+48x+36=24-4x$

Step 6.2:

Combine like terms.

$16x^2+52x+12=0$

Step 6.3:

Factor the quadratic equation.

$4(4x^2+13x+3)=0$

Step 6.4:

Find the roots of the equation.

$x=-\frac{1}{4},-3$

Step 7:

Check the solutions in the original equation to ensure they are valid.

$x=-3$

Knowledge Notes:

To solve the given rational equation involving square roots, we use the following mathematical concepts and techniques:

1. Squaring both sides: This is used to eliminate square roots. However, we must be cautious as squaring can introduce extraneous solutions.

2. Simplifying expressions: This involves using algebraic rules to rewrite expressions in a simpler form.

3. Distributive property (FOIL method): This is used to expand binomials. For example, $(a+b{)}^2=a^2+2ab+b^2$.

4. Combining like terms: This is the process of adding or subtracting terms that have the same variable raised to the same power.

5. Factoring quadratic equations: This involves rewriting the quadratic equation in a product form, if possible, to find its roots.

6. Checking solutions: After finding potential solutions, we substitute them back into the original equation to verify that they do not result in undefined expressions or contradictions.

7. Extraneous solutions: These are solutions that arise from the process of solving the equation but do not satisfy the original equation. They must be identified and excluded from the final answer.