Problem 17 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$30 y^{2}+23 y-40=0$$

Step-by-Step Solution

Answer

The solutions are \( y = \frac{5}{6} \) and \( y = \frac{-8}{5} \).

Step 1: Identify the coefficients

The given equation is \(30y^2 + 23y - 40 = 0\). The quadratic equation is in the form \(ay^2 + by + c = 0\). Here, identify the coefficients as \(a = 30\), \(b = 23\), and \(c = -40\).

Step 2: Recall the quadratic formula

The quadratic formula to find the roots of the equation \(ax^2 + bx + c = 0\) is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this formula to find the roots of the equation.

Step 3: Calculate the discriminant

The discriminant \(\Delta\) is given by \(b^2 - 4ac\). Substitute the values to find \(b^2 - 4ac = 23^2 - 4 \cdot 30 \cdot (-40)\). Calculate this to find \(\Delta = 529 + 4800 = 5329\).

Step 4: Determine the nature of the roots

Since the discriminant \(\Delta = 5329\) is greater than zero, the quadratic equation will have two distinct real roots.

Step 5: Substitute in the quadratic formula

Use the quadratic formula: \( y = \frac{-23 \pm \sqrt{5329}}{60} \). Calculate \(\sqrt{5329}\) which is 73.

Step 6: Calculate each root

There are two roots to find: 1. \( y_1 = \frac{-23 + 73}{60} = \frac{50}{60} = \frac{5}{6}\)2. \( y_2 = \frac{-23 - 73}{60} = \frac{-96}{60} = \frac{-8}{5}\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It allows us to find the roots, or solutions, of these equations efficiently. The formula is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This means to find the values of \(x\) that satisfy the equation, you substitute the coefficients \(a\), \(b\), and \(c\) into the formula. The symbol "\(\pm\)" indicates that there are two solutions to the quadratic equation. In simpler terms, the formula gives us two possible "x" values that will make the equation true. By finding these values, we determine where the quadratic graph (parabola) intersects the x-axis if real solutions exist.

Discriminant

The discriminant is a key part of the quadratic formula, nestled inside the square root. It is given by \(b^2 - 4ac\). The value of the discriminant helps determine the nature and number of roots an equation has:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), there is one real root (also called a repeated or double root).
• If \(\Delta < 0\), there are no real roots; the roots are complex or imaginary.

In the example provided, calculating the discriminant \(23^2 - 4 \times 30 \times (-40)\) resulted in a positive value of 5329. This confirms that the quadratic equation has two distinct real roots, as a positive discriminant signifies.

Real Roots

Real roots are the solutions to a quadratic equation that can be represented on the real number line. If a quadratic equation has real roots, it means that the graph of the equation intersects the x-axis at real number points.

• Real roots can be distinct, meaning the graph intersects the x-axis at two separate points.
• Or they can coincide when they are the same number, resulting in a graph that just touches the x-axis at one point.

In our solution, because the discriminant is greater than zero, the equation has two distinct real roots. This is further confirmed by using the quadratic formula to find \( y_1 = \frac{5}{6} \) and \( y_2 = \frac{-8}{5} \), these being the points where the graph crosses the x-axis.

Coefficients

In a quadratic equation like \(30y^2 + 23y - 40 = 0\), the terms \(a\), \(b\), and \(c\) are referred to as the coefficients. Each of these plays a crucial role in determining the shape and position of the quadratic equation's graph:

• \(a\) is the coefficient of the quadratic term \(y^2\). It influences the direction and width of the parabola. If \(a\) is positive, the parabola opens upwards. If negative, it opens downwards.
• \(b\) is the coefficient of the linear term \(y\), affecting the position of the vertex and axis of symmetry.
• \(c\) is the constant term, which moves the parabola up or down on the graph.

In this exercise, \(a = 30\), \(b = 23\), and \(c = -40\) precisely determine the nature of the quadratic equation, its graph, and the use of the quadratic formula to find the roots.