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Magazine Chapter 5: Problem 44
Solve. $$3 x^{2}-8 x+4=0$$
Short Answer
Expert verified
The solutions are \ x = 2 \ and \ x = \frac{2}{3}\.
Step by step solution
01
- Identify coefficients
First, identify the coefficients in the quadratic equation. For the equation \(3x^2 - 8x + 4 = 0\) , the coefficients are: \(a = 3, \ b = -8, \ c = 4\).
02
- Apply the quadratic formula
Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \ a, \ b, and \ c \ into the formula: \(x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot 4}}{2 \cdot 3}\).
03
- Simplify the expression inside the square root
Calculate the discriminant \(b^2 - 4ac\): \((-8)^2 - 4 \cdot 3 \cdot 4 = 64 - 48 = 16\).
04
- Calculate the square root
Simplify \ \sqrt{16}\ : \ \sqrt{16} = 4\. Thus, our equation becomes \(x = \frac{8 \pm 4}{6}\).
05
- Solve for the roots
Solve for the two possible values of \ x \: \(x_1 = \frac{8 + 4}{6} = 2\) and \(x_2 = \frac{8 - 4}{6} = \frac{2}{3}\). These are the solutions to the quadratic equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
To solve any quadratic equation of the form \(ax^2 + bx + c = 0\), we use the quadratic formula. The formula allows us to find the roots (solutions) of the equation. The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula might look complex at first, but each part has a specific role:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula might look complex at first, but each part has a specific role:
- The term \(-b\) helps in reversing the sign of the coefficient \(b\).
- The term \(\pm \sqrt{b^2 - 4ac}\) accounts for both possible solutions, adding and subtracting the square root part.
- Finally, the entire expression is divided by \(2a\), the doubled coefficient of \(x^2\).
Discriminant
The discriminant is a crucial part of the quadratic formula and is found inside the square root part, \(\sqrt{b^2 - 4ac}\). The discriminant itself is:\[\Delta = b^2 - 4ac\]
The value of the discriminant gives us important information about the nature of the roots:
The value of the discriminant gives us important information about the nature of the roots:
- If \(\Delta > 0\), the equation has two distinct real roots.
- If \(\Delta = 0\), the equation has exactly one real root (a repeated root).
- If \(\Delta < 0\), the equation has two complex roots (no real solutions).
Roots of Quadratic Equation
The roots or solutions of a quadratic equation are the values of \(x\) that satisfy the equation. Using the quadratic formula, we substitute the coefficients and solve for \(x\). In our case:
\[x = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 3}\] This simplifies to:
\(x = \frac{8 \pm 4}{6}\)
Which further breaks down to two separate solutions:
\[x = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 3}\] This simplifies to:
\(x = \frac{8 \pm 4}{6}\)
Which further breaks down to two separate solutions:
- \(x_1 = \frac{8 + 4}{6} = 2\)
- \(x_2 = \frac{8 - 4}{6} = \frac{2}{3}\)
Coefficients of Quadratic Equations
In any quadratic equation of the form \(ax^2 + bx + c = 0\), the letters \(a\), \(b\), and \(c\) are called coefficients. They represent specific numbers that define different aspects of the equation:
- \(a\) is the coefficient of \(x^2\). It determines the direction and width of the parabola. For \(3x^2 - 8x + 4 = 0\), \(a = 3\).
- \(b\) is the coefficient of \(x\). It affects the position of the parabola along the x-axis. Here, \(b = -8\).
- \(c\) is the constant term. It affects the vertical position of the parabola. In this equation, \(c = 4\).
