Problem 60 - Calculus: One and Several Variables

Question

Find the real roots of the equation. \(2 x^{2}-5 x-3=0\).

Step-by-Step Solution

Answer

The real roots of the quadratic equation \(2x^{2}-5x-3 = 0\) are 3 and -0.5.

Step 1: Identifying coefficients

Identify the coefficients in the quadratic formula, which takes the general form of \(ax^{2} + bx + c = 0\). In our given equation \(2x^{2} - 5x - 3 = 0\), 'a' is 2, 'b' is -5, and 'c' is -3.

Step 2: Calculating the Discriminant

Calculate the Discriminant, which is \(b^{2} - 4ac\). In this case, the Discriminant is \((-5)^{2} - 4*2*(-3) = 25 + 24 = 49\)

Step 3: Finding the Real Roots

Now use the quadratic formula \(-b \pm \sqrt{b^{2}-4ac} \over 2a\) to determine the roots of the equation. This gives two possible solutions for x: \(x1= \frac{-(-5) + \sqrt{49}}{2*2}=3\) and \(x2= \frac{-(-5) - \sqrt{49}}{2*2}=-0.5\)

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool for solving quadratic equations, which are any equations that can be written in the standard form of \(ax^2 + bx + c = 0\). When you encounter a quadratic equation and need to find the variable \(x\), the quadratic formula provides a straightforward solution:

\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]
Here, the \(\pm\) symbol indicates that there are generally two solutions to a quadratic equation, as the square root function can yield both a positive and a negative result. The \(a\), \(b\), and \(c\) in the formula represent the coefficients of the equation, which are the numerical factors that appear alongside the variable \(x\). By simply substituting the values of these coefficients into the quadratic formula, you can efficiently solve for the real roots of the equation.

Discriminant of Quadratic

The discriminant of a quadratic equation is a key component that determines the nature and number of roots the equation possesses. It is derived from the quadratic formula and represented by the symbol \(\Delta\), which corresponds to the expression within the square root:

\[\Delta = b^2 - 4ac\]
The discriminant tells us whether an equation has two real and distinct solutions (when \(\Delta > 0\)), one real and repeated solution (when \(\Delta = 0\)), or two complex solutions (when \(\Delta < 0\)). When calculating the discriminant of the quadratic equation \(2x^2 - 5x - 3 = 0\), we find that the discriminant is 49, which is a positive number. Thus, this ensures that the equation has two real and distinct roots.

Real Roots of Equation

Finding the real roots of a quadratic equation means determining the x-values where the graph of the quadratic function intersects the x-axis. These roots are the solutions to the equation \(ax^2 + bx + c = 0\) and can be obtained using the quadratic formula. If the discriminant \(\Delta\) is non-negative, the equation has real roots. In the example of the equation \(2x^2 - 5x - 3 = 0\), the roots are calculated to be \(3\) and \(-0.5\) using the quadratic formula. These values mean that you can graphically visualize the equation as a parabola crossing the x-axis at these two points, thus representing the real roots of the equation.

Coefficients of Quadratic

The coefficients of a quadratic equation are the numerical values that multiply the variable \(x\) in its various degrees. The standard form of a quadratic equation is written as \(ax^2 + bx + c = 0\), where:

• \(a\) is the coefficient of the term with \(x^2\)
• \(b\) is the coefficient of the term with \(x\)
• \(c\) is the constant term with no \(x\)

In the exercise provided, the coefficient \(a\) is 2, \(b\) is -5, and \(c\) is -3. These coefficients are crucial as they not only define the curvature and the orientation of the parabola when graphed but also are directly plugged into the quadratic formula and discriminant calculation to find the roots of the equation.