Problem 34 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ 3 x^{2}+7 x+4=0 $$

Step-by-Step Solution

Answer

The real solutions are \(x = -1\) and \(x = -\frac{4}{3}\).

Step 1: Identify the Quadratic Formula

The quadratic formula is used to find solutions to a quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this exercise, \(a = 3\), \(b = 7\), and \(c = 4\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation is \(b^2 - 4ac\). It determines the nature of the roots. Here, compute \(7^2 - 4 \times 3 \times 4 = 49 - 48 = 1\). Since the discriminant is positive, there are two distinct real solutions.

Step 3: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions. Substitute \(b = 7\), the discriminant \(\sqrt{1} = 1\), and \(a = 3\) into the formula: \(x = \frac{-7 \pm 1}{6}\).

Step 4: Solve for Two Roots

Calculate the two possible solutions from the formula: \(x_1 = \frac{-7 + 1}{6} = \frac{-6}{6} = -1\) and \(x_2 = \frac{-7 - 1}{6} = \frac{-8}{6} = -\frac{4}{3}\).

Step 5: Present the Real Solutions

The equation has two real solutions: \(x = -1\) and \(x = -\frac{4}{3}\). These are the values of \(x\) that satisfy the initial equation.

Key Concepts

Quadratic Formula

The quadratic equation is a key tool for solving quadratic equations, which have the form \( ax^2 + bx + c = 0 \). This formula helps in finding the values of \( x \) that satisfy the equation. To use the quadratic formula, you need to identify the coefficients \( a \), \( b \), and \( c \). Here, they are 3, 7, and 4 respectively. The formula is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula may look complex, but it's simple once you break it down:- It uses the coefficients from the equation to substitute the values for \( a \), \( b \), and \( c \).- The \( \pm \) symbol indicates there will be two solutions. Thus, it captures both the addition and subtraction of the square root value.Using this formula provides a systematic way to determine the solutions of any quadratic equation.

Discriminant

The discriminant is a part of the quadratic formula and is crucial in understanding the nature of the solutions of a quadratic equation. It is the expression \( b^2 - 4ac \) found under the square root in the quadratic formula.In this exercise, the discriminant is calculated as follows: \[ 7^2 - 4 \times 3 \times 4 = 49 - 48 = 1 \] What does this mean for the roots?- A positive discriminant implies two distinct real solutions.- A discriminant of zero means there is exactly one real solution (a repeated root).- A negative discriminant indicates there are no real solutions, but two complex ones.Here, since the discriminant is 1, we have two distinct real solutions.

Real Solutions

Real solutions of a quadratic equation arise when the discriminant is zero or positive. They are the actual values of \( x \) that satisfy the original quadratic equation. To find these real solutions, you apply the quadratic formula with the values calculated from the discriminant.For the equation \( 3x^2 + 7x + 4 = 0 \), after calculating the discriminant as 1, the real solutions are determined by: \[ x = \frac{-7 \pm 1}{6} \]Following through:- Adding inside the formula gives the first solution \( x_1 = \frac{-6}{6} = -1 \)- Subtracting gives the second solution \( x_2 = \frac{-8}{6} = -\frac{4}{3} \)Thus, the real solutions are \( x = -1 \) and \( x = -\frac{4}{3} \). They are the actual values that make the equation equal to zero.

Roots of Quadratic Equations

Roots or solutions of a quadratic equation are essentially the values of \( x \) at which the equation equals zero. For quadratic equations, understanding the roots is vital, as they signify the points where the parabola represented by the equation intersects the \( x \)-axis. These roots can be classified based on the discriminant:- **Two Distinct Real Roots:** This occurs when the discriminant is positive. The parabola crosses the \( x \)-axis at two different points.- **One Real Root:** Known as a repeated or double root, this happens when the discriminant is zero. The parabola just touches the \( x \)-axis.- **Two Complex Roots:** If the discriminant is negative, the parabola does not intersect the \( x \)-axis, indicating there are no real roots, just complex.In our case, both roots \( x = -1 \) and \( x = -\frac{4}{3} \) are real, showing where the curve crosses the \( x \)-axis.