Solve for x x- square root of x=12

Solution:

Step:1

Isolate the square root term by subtracting $x$ from both sides: $- \sqrt{x} = 12 - x$

Step:2

Square both sides to eliminate the square root: $(- \sqrt{x})^{2} = (12 - x)^{2}$

Step:3

Expand and simplify both sides of the equation.

Step:3.1

Express $\sqrt{x}$ as $x^{\frac{1}{2}}$: $(- x^{\frac{1}{2}})^{2} = (12 - x)^{2}$

Step:3.2

Simplify the left side by squaring the terms.

Step:3.2.1

Square $- x^{\frac{1}{2}}$: $(-1)^{2} (x^{\frac{1}{2}})^{2} = (12 - x)^{2}$

Step:3.2.1.1

Square $-1$: $1 (x^{\frac{1}{2}})^{2} = (12 - x)^{2}$

Step:3.2.1.2

Multiply the squared root by $1$: $(x^{\frac{1}{2}})^{2} = (12 - x)^{2}$

Step:3.2.1.3

Apply the power rule to the exponent: $x^{\frac{1}{2} \cdot 2} = (12 - x)^{2}$

Step:3.2.1.4

Simplify the exponent: $x^{1} = (12 - x)^{2}$

Step:3.2.1.5

Resulting in: $x = (12 - x)^{2}$

Step:3.3

Simplify the right side by expanding the binomial.

Step:3.3.1

Expand $(12 - x)^{2}$: $x = (12 - x)(12 - x)$

Step:3.3.1.1

Use the FOIL method to expand: $x = 12(12 - x) - x(12 - x)$

Step:3.3.1.2

Distribute: $x = 12 \cdot 12 - 12x - x \cdot 12 + x \cdot x$

Step:3.3.1.3

Combine like terms: $x = 144 - 24x + x^{2}$

Step:4

Find the value(s) of $x$.

Step:4.1

Rearrange the equation: $x^{2} - 24x + 144 = x$

Step:4.2

Move all terms with $x$ to one side: $x^{2} - 25x + 144 = 0$

Step:4.3

Factor the quadratic equation.

Step:4.3.1

Find two numbers that multiply to $144$ and add to $-25$: $-16$ and $-9$

Step:4.3.2

Write the factors: $(x - 16)(x - 9) = 0$

Step:4.4

Apply the zero-product property: $x - 16 = 0$ or $x - 9 = 0$

Step:4.5

Solve the first equation: $x = 16$

Step:4.6

Solve the second equation: $x = 9$

Step:4.7

The potential solutions are $x = 16$ or $x = 9$

Step:5

Verify which solution satisfies the original equation: $x = 16$

Knowledge Notes:

1. Square Roots and Exponents: The square root of a number $x$ can be written as $x^{\frac{1}{2}}$. Squaring the square root, $(\sqrt{x})^{2}$, returns the original number $x$.

2. Squaring Both Sides: When dealing with equations that contain square roots, it's common to square both sides to eliminate the radical. However, this can introduce extraneous solutions, so it's important to check the solutions in the original equation.

3. Expanding Binomials: The square of a binomial, $(a - b)^{2}$, can be expanded using the FOIL method (First, Outer, Inner, Last) to get $a^{2} - 2ab + b^{2}$.

4. Zero-Product Property: If a product of factors equals zero, at least one of the factors must be zero. This property is used to solve quadratic equations that have been factored into the form $(x - a)(x - b) = 0$.

5. Quadratic Equations: A quadratic equation is an equation of the form $ax^{2} + bx + c = 0$. It can often be solved by factoring, completing the square, or using the quadratic formula.

6. Extraneous Solutions: When solving equations, especially those involving radicals or rational expressions, it's possible to obtain solutions that do not satisfy the original equation. These are called extraneous solutions and must be discarded.

7. Verification: After solving an equation, it's important to substitute the solutions back into the original equation to verify their validity.