Solve for a a = square root of 4a+5
The problem you've presented requires determining the value of the variable "a" that satisfies the equation in which "a" is equal to the square root of "4a + 5". To solve this problem, one would typically isolate the variable on one side of the equation and solve for it, keeping in mind the properties of square roots and squares, and ensuring that any solutions found are checked for extraneous solutions that may arise from squaring both sides of the equation.
$a = \sqrt{4 a + 5}$
Answer
Solution:
Step 1:
Move the radical to the left side by rewriting the equation as $\sqrt{4a + 5} = a$.
Step 2:
Eliminate the square root by squaring both sides: $(\sqrt{4a + 5})^2 = a^2$.
Step 3:
Simplify the equation.
Step 3.1:
Express the square root as a power: $(4a + 5)^{\frac{1}{2}}$.
Step 3.2:
Simplify the left side by squaring the power.
Step 3.2.1:
Apply exponent rules: $(4a + 5)^{\frac{1}{2} \cdot 2}$.
Step 3.2.1.1:
Multiply the exponents: $(4a + 5)^{1}$.
Step 3.2.1.2:
Simplify the expression: $4a + 5 = a^2$.
Step 4:
Solve for $a$ by rearranging the equation.
Step 4.1:
Subtract $a^2$ from both sides: $4a + 5 - a^2 = 0$.
Step 4.2:
Factor the quadratic equation.
Step 4.2.1:
Reorder and factor out $-1$: $-a^2 + 4a + 5 = 0$.
Step 4.2.2:
Factor the quadratic: $(a - 5)(a + 1) = 0$.
Step 4.3:
Set each factor equal to zero to find possible solutions: $a - 5 = 0$ and $a + 1 = 0$.
Step 4.4:
Solve for $a$ from $a - 5 = 0$: $a = 5$.
Step 4.5:
Solve for $a$ from $a + 1 = 0$: $a = -1$.
Step 4.6:
Combine the solutions: $a = 5, -1$.
Step 5:
Verify which solutions satisfy the original equation: $a = 5$.
Knowledge Notes:
To solve the equation $a = \sqrt{4a + 5}$, we follow a systematic approach:
Square Roots and Powers: We use the property that squaring a square root will eliminate the radical, i.e., $(\sqrt{x})^2 = x$.
Simplifying Equations: After squaring both sides, we simplify the equation by applying the power rule $(a^m)^n = a^{mn}$.
Quadratic Equations: The resulting equation is a quadratic, which we can solve by factoring or using the quadratic formula. In this case, we factor the equation.
Zero Product Property: If a product of two factors equals zero, then at least one of the factors must be zero. This property allows us to set each factor equal to zero to find the solutions.
Verification of Solutions: Not all solutions found algebraically will satisfy the original equation, especially when dealing with square roots (which imply non-negative results). We must substitute the solutions back into the original equation to verify their validity.
Exclusion of Extraneous Solutions: In equations involving radicals, it's possible to obtain extraneous solutions that do not satisfy the original equation. These must be excluded from the final answer.
