Solve X-√x=12 Using The Formula
Introduction
The equation x - √x = 12 is a quadratic equation that involves a square root term. Solving this equation requires a combination of algebraic manipulations and the use of the quadratic formula. In this article, we will provide a step-by-step guide on how to solve the equation x - √x = 12 using the quadratic formula.
Understanding the Equation
The given equation is x - √x = 12. To solve this equation, we need to isolate the square root term. We can do this by adding √x to both sides of the equation, which gives us:
x - √x + √x = 12 + √x
Simplifying the left-hand side of the equation, we get:
x = 12 + √x
Rearranging the Equation
To make it easier to solve the equation, we can rearrange it to isolate the square root term. We can do this by subtracting 12 from both sides of the equation, which gives us:
x - 12 = √x
Squaring Both Sides
To eliminate the square root term, we can square both sides of the equation. This gives us:
(x - 12)^2 = (√x)^2
Expanding the left-hand side of the equation, we get:
x^2 - 24x + 144 = x
Rearranging the Equation
To make it easier to solve the equation, we can rearrange it to get all the terms on one side. We can do this by subtracting x from both sides of the equation, which gives us:
x^2 - 25x + 144 = 0
Solving the Quadratic Equation
The equation x^2 - 25x + 144 = 0 is a quadratic equation that can be solved using the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -25, and c = 144. Plugging these values into the quadratic formula, we get:
x = (25 ± √((-25)^2 - 4(1)(144))) / 2(1)
Simplifying the expression under the square root, we get:
x = (25 ± √(625 - 576)) / 2
x = (25 ± √49) / 2
x = (25 ± 7) / 2
Finding the Solutions
We can now find the two possible solutions for x by plugging in the values of ±7 into the equation. We get:
x = (25 + 7) / 2 = 32 / 2 = 16
x = (25 - 7) / 2 = 18 / 2 = 9
Checking the Solutions
To check if these solutions are valid, we can plug them back into the original equation. We get:
x - √x = 16 - √16 = 16 - 4 = 12
x - √x = 9 - √9 = 9 - 3 = 6
Since the first solution satisfies the original equation, we can conclude that x = 16 is a valid solution. However, the second solution does not satisfy the original equation, so we can conclude that x = 9 is not a valid solution.
Conclusion
In this article, we provided a step-by-step guide on how to solve the equation x - √x = 12 using the quadratic formula. We rearranged the equation to isolate the square root term, squared both sides to eliminate the square root term, and then solved the resulting quadratic equation using the quadratic formula. We found two possible solutions for x, but only one of them satisfied the original equation.
Q: What is the first step in solving the equation x - √x = 12?
A: The first step in solving the equation x - √x = 12 is to add √x to both sides of the equation to isolate the square root term.
Q: How do I simplify the left-hand side of the equation x - √x + √x = 12 + √x?
A: To simplify the left-hand side of the equation, we can combine like terms. In this case, the like terms are the √x terms, which cancel each other out. This leaves us with x = 12 + √x.
Q: Why do I need to rearrange the equation x = 12 + √x to isolate the square root term?
A: We need to rearrange the equation to isolate the square root term because it makes it easier to solve the equation. By subtracting 12 from both sides of the equation, we can get the equation in a form where we can eliminate the square root term.
Q: What is the next step after rearranging the equation x - 12 = √x?
A: The next step is to square both sides of the equation to eliminate the square root term. This gives us (x - 12)^2 = (√x)^2.
Q: How do I expand the left-hand side of the equation (x - 12)^2 = (√x)^2?
A: To expand the left-hand side of the equation, we can use the formula (a - b)^2 = a^2 - 2ab + b^2. In this case, a = x and b = 12. This gives us x^2 - 24x + 144 = (√x)^2.
Q: What is the next step after expanding the left-hand side of the equation x^2 - 24x + 144 = (√x)^2?
A: The next step is to simplify the right-hand side of the equation by squaring the square root term. This gives us x^2 - 24x + 144 = x.
Q: How do I rearrange the equation x^2 - 24x + 144 = x to get all the terms on one side?
A: To rearrange the equation, we can subtract x from both sides of the equation. This gives us x^2 - 25x + 144 = 0.
Q: What is the quadratic formula, and how do I use it to solve the equation x^2 - 25x + 144 = 0?
A: The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 1, b = -25, and c = 144. Plugging these values into the quadratic formula, we get x = (25 ± √((-25)^2 - 4(1)(144))) / 2(1).
Q: How do I simplify the expression under the square root in the quadratic formula x = (25 ± √((-25)^2 - 4(1)(144))) / 2(1)?
A: To simplify the expression under the square root, we can calculate the value of (-25)^2, which is 625, and the value of 4(1)(144), which is 576. Plugging these values into the expression, we get x = (25 ± √(625 - 576)) / 2.
Q: What is the next step after simplifying the expression under the square root in the quadratic formula x = (25 ± √(625 - 576)) / 2?
A: The next step is to simplify the expression under the square root by calculating the value of 625 - 576, which is 49. This gives us x = (25 ± √49) / 2.
Q: How do I simplify the expression x = (25 ± √49) / 2 further?
A: To simplify the expression, we can calculate the value of √49, which is 7. This gives us x = (25 ± 7) / 2.
Q: What are the two possible solutions for x in the equation x = (25 ± 7) / 2?
A: The two possible solutions for x are x = (25 + 7) / 2 = 32 / 2 = 16 and x = (25 - 7) / 2 = 18 / 2 = 9.
Q: How do I check if the solutions x = 16 and x = 9 satisfy the original equation x - √x = 12?
A: To check if the solutions satisfy the original equation, we can plug them back into the equation. We get x - √x = 16 - √16 = 16 - 4 = 12 and x - √x = 9 - √9 = 9 - 3 = 6.
Q: Which of the two solutions x = 16 and x = 9 satisfies the original equation x - √x = 12?
A: The solution x = 16 satisfies the original equation x - √x = 12, while the solution x = 9 does not.
Q: What is the final answer to the equation x - √x = 12?
A: The final answer to the equation x - √x = 12 is x = 16.