Solve for x square root of 9x-3 = square root of x+14
The problem is asking for the solution to an equation where the square root of a linear expression in x (9x - 3) is set equal to the square root of another expression in x (x + 14). You need to find the value(s) of x that would satisfy this equation when both sides of the equation are considered equal. Essentially, you're looking for the intersection of the two functions defined by the square roots of their respective expressions.
$\sqrt{9 x} - 3 = \sqrt{x + 14}$
Answer
Solution:
Step:1
Shift $3$ to the opposite side of the equation from both sides. $\sqrt{9x} = \sqrt{x + 14} + 3$
Step:2
Square both sides to eliminate the square root on the left. $(\sqrt{9x})^2 = (\sqrt{x + 14} + 3)^2$
Step:3
Expand and simplify both sides of the equation.
Step:3.1
Express $\sqrt{9x}$ using exponent notation. $(9x)^{\frac{1}{2}}$
Step:3.2
Simplify the left side by applying exponent rules.
Step:3.2.1
Apply the power of a power rule. $(9x)^{\frac{1}{2} \cdot 2}$
Step:3.2.2
Simplify by canceling out the exponent of $2$. $9x = (\sqrt{x + 14} + 3)^2$
Step:3.3
Expand and simplify the right side.
Step:3.3.1
Expand the squared term on the right. $(\sqrt{x + 14} + 3)(\sqrt{x + 14} + 3)$
Step:3.3.2
Use the distributive property (FOIL) to expand. $9x = x + 14 + 6\sqrt{x + 14} + 9$
Step:4
Isolate the term with the square root. $6\sqrt{x + 14} = 9x - x - 23$
Step:5
Square both sides again to remove the square root. $(6\sqrt{x + 14})^2 = (9x - x - 23)^2$
Step:6
Simplify both sides of the squared equation.
Step:6.1
Express $\sqrt{x + 14}$ using exponent notation. $(x + 14)^{\frac{1}{2}}$
Step:6.2
Simplify the left side by applying exponent rules.
Step:6.2.1
Apply the power of a power rule. $36(x + 14) = (8x - 23)^2$
Step:7
Solve for $x$ by setting up and solving a quadratic equation.
Step:7.1
Move all terms to one side to form a quadratic equation. $64x^2 - 404x + 529 = 36x + 504$
Step:7.2
Combine like terms and set the equation to zero. $64x^2 - 440x + 25 = 0$
Step:7.3
Factor the quadratic equation. $(16x - 1)(4x - 25) = 0$
Step:8
Find the roots of the equation by setting each factor equal to zero.
Step:8.1
Solve $16x - 1 = 0$ for $x$. $x = \frac{1}{16}$
Step:8.2
Solve $4x - 25 = 0$ for $x$. $x = \frac{25}{4}$
Step:9
Check the solutions in the original equation to ensure they are valid. $x = \frac{25}{4}$
Step:10
Present the solution in various forms.
Exact Form: $x = \frac{25}{4}$ Decimal Form: $x = 6.25$ Mixed Number Form: $x = 6 \frac{1}{4}$
Knowledge Notes:
The problem involves solving an equation with square roots. The key steps in solving such equations include:
Isolating the square root on one side of the equation.
Squaring both sides of the equation to eliminate the square root.
Simplifying the resulting equation, which may involve expanding binomials and combining like terms.
Factoring and solving the resulting quadratic equation, if necessary.
Checking the solutions in the original equation to exclude extraneous solutions that may arise from squaring both sides.
Relevant algebraic rules used in the process include:
The power of a power rule: $(a^{m})^{n} = a^{mn}$
The distributive property (FOIL Method) for expanding binomials: $(a + b)(c + d) = ac + ad + bc + bd$
Factoring by grouping, which involves combining terms with common factors and factoring them out.
The zero-product property: If $ab = 0$, then either $a = 0$ or $b = 0$ or both.
It is important to note that when squaring both sides of an equation, extra solutions called extraneous solutions may be introduced. These solutions do not satisfy the original equation and must be checked for and discarded if necessary.
