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Magazine Chapter 11: Problem 8
Solve. $$ 3 x^{2}-7 x+2=0 $$
Short Answer
Expert verified
The solutions are x = 2 and x = 1/3.
Step by step solution
01
Identify coefficients
Identify the coefficients in the quadratic equation in the form of ax^2 + bx + c = 0. For the equation 3x^2 - 7x + 2 = 0, the coefficients are a = 3, b = -7, and c = 2.
02
Apply the quadratic formula
The quadratic formula is given by \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]. Substitute the identified coefficients into the formula.
03
Substitute values into the formula
Substitute a = 3, b = -7, and c = 2 into the quadratic formula: \[x = \frac{{-(-7) \pm \sqrt{{(-7)^2 - 4(3)(2)}}}}{{2(3)}}\].
04
Simplify under the square root
Simplify the expression inside the square root: \[x = \frac{{7 \pm \sqrt{{49 - 24}}}}{{6}}\].
05
Compute the square root and solve
Continue simplifying: \[x = \frac{{7 \pm \sqrt{{25}}}}{{6}}\]. Then, calculate the square root of 25: \[x = \frac{{7 \pm 5}}{{6}}\].
06
Find both solutions
Calculate both possible solutions: \[x = \frac{{7 + 5}}{{6}} = 2\] and \[x = \frac{{7 - 5}}{{6}} = \frac{{1}}{{3}}\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
quadratic formula
When solving quadratic equations, the quadratic formula is a crucial tool. It's used to find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is expressed as: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]Here \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. This formula helps us find where the graph of the quadratic equation crosses the x-axis, or simply the values of \(x\) that satisfy the equation.
coefficients
In any quadratic equation \(ax^2 + bx + c = 0\), the coefficients are the constants that multiply each term:
- \(a\): The coefficient of \(x^2\)
- \(b\): The coefficient of \(x\)
- \(c\): The constant term
simplifying expressions
Simplifying expressions is a key step in solving quadratic equations using the quadratic formula. Let's break it down for the equation \(3x^2 - 7x + 2 = 0\): 1. Start with the quadratic formula: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]2. Substitute the coefficients (\(a = 3\), \(b = -7\), \(c = 2\)) into the formula:\[x = \frac{{-(-7) \pm \sqrt{{(-7)^2 - 4(3)(2)}}}}{{2(3)}}\]3. Simplify under the square root: \[x = \frac{{7 \pm \sqrt{{49 - 24}}}}{{6}}\]4. After combining and simplifying under the root: \[x = \frac{{7 \pm \sqrt{{25}}}}{{6}}\]The expression under the root must be simplified accurately to avoid errors.
finding roots
Finding the roots of a quadratic equation using the quadratic formula involves calculating the final values after simplification. For the equation \(3x^2 - 7x + 2 = 0\), we continue from previous simplifying steps: \[x = \frac{{7 \pm \sqrt{{25}}}}{{6}}\]Since \(\sqrt{25} = 5 \), we have: \[x = \frac{{7 \pm 5}}{{6}}\]This results in two solutions:
- \(x = \frac{{7 + 5}}{{6}} = 2\)
- \(x = \frac{{7 - 5}}{{6}} = \frac{1}{3}\)
