Factor 12x-5 square root of x-2

Solution:

Step:1

Convert $\sqrt{x}$ to $x^{\frac{1}{2}}$ using the property $\sqrt[n]{a^x}=a^{\frac{x}{n}}$. The expression becomes $12x-5x^{\frac{1}{2}}-2$.

Step:2

Express $x$ as ${\left(x^{\frac{1}{2}}\right)}^2$. The expression now reads $12{\left(x^{\frac{1}{2}}\right)}^2-5x^{\frac{1}{2}}-2$.

Step:3

Introduce a substitution where $u=x^{\frac{1}{2}}$. Replace $x^{\frac{1}{2}}$ with $u$ to get $12u^2-5u-2$.

Step:4

Begin the process of factoring by grouping.

Step:4.1

For the quadratic form $a{x}^2+bx+c$, split the middle term into two terms that multiply to $a\cdot c=12\cdot -2=-24$ and add up to $b=-5$.

Step:4.1.1

Extract $-5$ from $-5u$. The expression is now $12u^2-5(u)-2$.

Step:4.1.2

Decompose $-5$ into $3$ and $-8$. The expression updates to $12u^2+(3-8)u-2$.

Step:4.1.3

Employ the distributive property to combine like terms: $12u^2+3u-8u-2$.

Step:4.2

Extract the greatest common factor (GCF) from each binomial group.

Step:4.2.1

Pair the first two and the last two terms: $(12u^2+3u)-(8u+2)$.

Step:4.2.2

Factor out the GCF from each pair: $3u(4u+1)-2(4u+1)$.

Step:4.3

Complete the factorization by taking out the common binomial factor $4u+1$: $(4u+1)(3u-2)$.

Step:5

Substitute back $u$ with $x^{\frac{1}{2}}$ to get the final factored form: $(4x^{\frac{1}{2}}+1)(3x^{\frac{1}{2}}-2)$.

Knowledge Notes:

The problem-solving process involves several key knowledge points:

1. Radical to Exponent Conversion: The expression $\sqrt[n]{a^x}=a^{\frac{x}{n}}$ is used to convert a square root to an exponent. This is a fundamental property of exponents and radicals.

2. Substitution Method: Introducing a new variable (like $u$) to simplify the expression is a common technique in algebra. It helps to transform a complex expression into a more familiar form, such as a quadratic equation.

3. Factoring by Grouping: This is a method used to factor polynomials that involve separating the polynomial into groups that can be factored separately and then combining the results.

4. Greatest Common Factor (GCF): The GCF is the largest factor that divides two or more numbers. Factoring out the GCF from each group simplifies the expression and is a crucial step in the factoring process.

5. Distributive Property: This property states that $a(b+c)=ab+ac$. It is used to expand and combine like terms in algebraic expressions.

6. Reverse Substitution: After factoring the expression with the substituted variable, it's important to replace the variable back with the original expression to complete the problem.

7. Quadratic Factoring: The process of factoring a quadratic expression involves finding two binomials that when multiplied together give the original quadratic. This is often done by finding two numbers that multiply to give the product of the coefficient of the squared term and the constant term, and add to give the coefficient of the linear term.