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Magazine Chapter 1: Problem 34
Find all real solutions of the equation. $$3 x^{2}+7 x+4=0$$
Short Answer
Expert verified
The solutions are \(x = -1\) and \(x = -\frac{4}{3}\).
Step by step solution
01
Identify the Structure of the Quadratic Equation
The given equation is \(3x^2 + 7x + 4 = 0\), which is quadratic in form. A general quadratic equation is expressed as \(ax^2 + bx + c = 0\), where in this case, \(a = 3\), \(b = 7\), and \(c = 4\).
02
Determine the Discriminant
The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). It helps determine the nature and number of the roots of the equation. Substitute the values: \(D = 7^2 - 4 \cdot 3 \cdot 4 = 49 - 48 = 1\). Since \(D > 0\), the equation has two distinct real roots.
03
Use the Quadratic Formula
The solutions of the quadratic equation are found using the quadratic formula: \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute \(a = 3\), \(b = 7\), and \(D = 1\) into the formula: \(x = \frac{-7 \pm \sqrt{1}}{6}\).
04
Calculate Each Root
Evaluate the expression to find the two roots. First root: \(x_1 = \frac{-7 + 1}{6} = \frac{-6}{6} = -1\). Second root: \(x_2 = \frac{-7 - 1}{6} = \frac{-8}{6} = -\frac{4}{3}\). Thus, the roots are \(x = -1\) and \(x = -\frac{4}{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Discriminant
When dealing with quadratic equations of the form \(ax^2 + bx + c = 0\), the discriminant is a key element to understanding the nature of the equation's solutions. The discriminant is calculated using the formula \(D = b^2 - 4ac\). This small part of the formula actually tells us a lot about the solutions:
- If \(D > 0\), the equation has two distinct real roots.
- If \(D = 0\), there is exactly one real root, meaning the roots are repeated or identical.
- If \(D < 0\), the equation has no real roots, but two complex roots instead.
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of a quadratic equation, \(ax^2 + bx + c = 0\). This formula is given by:\[x = \frac{-b \pm \sqrt{D}}{2a}\]The plus-minus symbol \(\pm\) indicates that there are potentially two solutions. The formula uses the already calculated discriminant \(D\) to determine these solutions. Here鈥檚 how each part contributes:
- \(-b\): This is the negation of the coefficient of \(x\), which forms part of both roots.
- \(\sqrt{D}\): The square root of the discriminant determines the difference between the two potential roots. If \(D\) is zero, the formula results in a single root.
- \(2a\): This denominator effectively scales the solutions based on the leading coefficient's value \(a\).
Real Roots
Real roots are the solutions to a quadratic equation where the solutions are real numbers, not involving any imaginary components. Understanding when quadratic equations yield such real roots is extremely important.
The number of real roots a quadratic equation has is largely predicted by the discriminant:
The number of real roots a quadratic equation has is largely predicted by the discriminant:
- When \(D > 0\), there are exactly two distinct real roots, as seen with our example equation, which produced \(x = -1\) and \(x = -\frac{4}{3}\).
- If \(D = 0\), there is one real root, signifying a 'double root' or a repeated root, indicating that the parabola touches the x-axis at one point.
- When \(D < 0\), no real roots exist, and instead, there are two complex roots, meaning the parabola doesn't intersect the x-axis at all.
