Solve the Rational Equation for x square root of 1-5x=1+ square root of 6-x
The given problem is asking to find the value of the variable x that satisfies the equation involving square roots. Specifically, the equation is \(\sqrt{1-5x} = 1 + \sqrt{6-x}\). To solve this equation, one must isolate the square root expressions, square both sides to eliminate the square roots, and then solve for x using algebraic methods. The solution process may involve simplifying the resulting equation, moving terms to one side to get a quadratic or linear equation, and checking the solution to ensure it does not produce extraneous roots due to the squaring process.
$\sqrt{1 - 5 x} = 1 + \sqrt{6 - x}$
Answer
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:
Squaring both sides: This is used to eliminate square roots. However, we must be cautious as squaring can introduce extraneous solutions.
Simplifying expressions: This involves using algebraic rules to rewrite expressions in a simpler form.
Distributive property (FOIL method): This is used to expand binomials. For example, $(a + b)^2 = a^2 + 2ab + b^2$.
Combining like terms: This is the process of adding or subtracting terms that have the same variable raised to the same power.
Factoring quadratic equations: This involves rewriting the quadratic equation in a product form, if possible, to find its roots.
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.
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.
