Problem 47 - Algebra and Trigonometry

Question

Find all real solutions of the equation. \(3 x^{2}+2 x+2=0\)

Step-by-Step Solution

Answer

The equation has no real solutions as its discriminant is negative.

Step 1: Recognize the Quadratic Equation

The equation given is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 3\), \(b = 2\), and \(c = 2\). We'll use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation is \(D = b^2 - 4ac\). Substitute \(a = 3\), \(b = 2\), and \(c = 2\) to find \(D\): \(D = 2^2 - 4 \cdot 3 \cdot 2 = 4 - 24 = -20\).

Step 3: Interpret the Discriminant

Since the discriminant is $-20$, which is less than zero, the equation has no real solutions. Real solutions only exist if the discriminant is greater than or equal to zero.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are polynomial equations of the form \(ax^2 + bx + c = 0\). This formula helps find the solutions for \(x\) by transforming the equation into a form that is easier to solve. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• The expression under the square root, \(b^2 - 4ac\), is crucial as it helps determine the nature of the roots.
• In this equation, \(a\), \(b\), and \(c\) are coefficients that you extract from your quadratic equation.

You simply plug these values into the formula to find the solutions.

Discriminant

The discriminant plays a key role in the quadratic formula, dictating the type and number of solutions available. Calculated as \(D = b^2 - 4ac\), it reveals critical details about the roots of the quadratic equation.

• If \(D > 0\), there are two distinct real solutions.
• When \(D = 0\), there is exactly one real solution, known as a repeated or double root.
• If \(D < 0\), no real solutions exist, only complex or imaginary solutions are possible.

In our exercise, the discriminant is \(-20\), indicating that the equation has no real roots.

Real Solutions

Real solutions to an equation are those solutions that can be expressed as real numbers. These can be any value on the number line, including integers, fractions, and irrationals.

• For a quadratic equation to have real solutions, the discriminant \(D\) must be non-negative.
• If \(D > 0\), the solutions are two different real numbers.
• With \(D = 0\), there is one real solution, which is considered a double root.

Since the exercise discriminant was negative, it tells us that there are no solutions within the set of real numbers.

Polynomial Equations

Polynomial equations include terms with powers of the variable, usually represented in the form \(anx^n + a_{n-1}x^{n-1} + ... + a1x + a0 = 0\). Quadratic equations are a special type of polynomial equation where the highest power of the variable \(x\) is 2.

• The standard form of a quadratic polynomial is \(ax^2 + bx + c\).
• The equation given in the exercise, \(3x^2 + 2x + 2 = 0\), is an example of a quadratic polynomial.

To solve polynomial equations, one can factorize them, use graphs, or apply formulas such as the quadratic formula when they are of degree 2. Understanding their structure aids in identifying suitable methods to find solutions.