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Magazine Chapter 3: Problem 28
In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 3+\sqrt{4 x-3}=2 x $$
Short Answer
Expert verified
The solution set is \( \{3\} \).
Step by step solution
01
Isolate the Square Root
Start by isolating the square root term on one side of the equation. The original equation is: \[ 3 + \sqrt{4x-3} = 2x \]Subtract 3 from both sides to get:\[ \sqrt{4x-3} = 2x - 3 \]
02
Square Both Sides
To eliminate the square root, square both sides of the equation:\[ (\sqrt{4x-3})^2 = (2x - 3)^2 \]This simplifies to:\[ 4x - 3 = 4x^2 - 12x + 9 \]
03
Rearrange and Simplify
Bring all terms to one side of the equation to form a quadratic equation:\[ 4x^2 - 12x + 9 - 4x + 3 = 0 \]Simplify it:\[ 4x^2 - 16x + 12 = 0 \]
04
Factor the Quadratic Equation
Factor the quadratic equation:2x can be taken common:\[ 4(x^2 - 4x + 3) = 0 \] This factors further into:\[ 4(x - 3)(x - 1) = 0 \]
05
Solve for x
Set each factor equal to zero and solve for \(x\):1. \(x - 3 = 0\) gives \(x = 3\)2. \(x - 1 = 0\) gives \(x = 1\)
06
Check Each Solution
Substitute \(x = 3\) back into the original equation:\[ 3 + \sqrt{4(3) - 3} = 2(3) \]\[ 3 + \sqrt{9} = 6 \]\[ 3 + 3 = 6 \]It checks out, so \(x = 3\) is a solution.Now check \(x = 1\):\[ 3 + \sqrt{4(1) - 3} = 2(1) \]\[ 3 + \sqrt{1} = 2 \]\[ 3 + 1 = 2 \]This is incorrect, so \(x = 1\) is not a solution.
07
Write the Solution Set
The only solution that satisfies the original equation is \(x = 3\). Therefore, the solution set is:\[ \{3\} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factorization
Factorization is a key step when solving quadratic equations. It transforms a complex polynomial into a product of simpler binomials. This makes finding solutions much easier. Consider the equation obtained after removing the square root from: \[ 4x^2 - 16x + 12 = 0 \] This quadratic equation can be factored by taking out the greatest common factor which in this case is 4: \[ 4(x^2 - 4x + 3) = 0 \]
- Next, focus on the trinomial \(x^2 - 4x + 3\). Look for two numbers that multiply to 3 and add to -4.
- These numbers are -3 and -1. Hence, the expression can be factored as \((x - 3)(x - 1)\).
Solution Sets
A solution set is a collection of values that satisfy an equation. After factorizing the quadratic equation into \((x - 3)(x - 1) = 0\), each bracket can be solved separately:
- Setting \(x - 3 = 0\), we find \(x = 3\).
- Likewise, setting \(x - 1 = 0\), gives \(x = 1\).
Checking Solutions
Checking solutions is an essential step in solving equations, as it determines which potential solutions are valid. Consider our findings of \(x = 3\) and \(x = 1\):
- To verify, substitute \(x = 3\) into the original equation \(3 + \sqrt{4x-3} = 2x\). It simplifies to \(6 = 6\), confirming \(x = 3\) is a solution.
- Substitute \(x = 1\), resulting in \(4 eq 2\). Thus, \(x = 1\) is not a solution.
Quadratic Expression
A quadratic expression is a polynomial of degree 2. It typically takes the form \(ax^2 + bx + c\). In our problem, we dealt with the quadratic \(4x^2 - 16x + 12\). Understanding the components of quadratic expressions is crucial:
- The term \(ax^2\) represents the quadratic or squared term, influencing the shape of the parabola.
- The term \(bx\) is the linear term, affecting the direction the parabola opens.
- The constant term \(c\) shifts the graph vertically.
