Problem 5 - Technical Mathematics with Calculus

Question

Solve by quadratic formula. Give your answers in decimal form to three significant digits. Check some by calculator. $$2 x^{2}-15 x+9=0$$

Step-by-Step Solution

Answer

x ≈ 0.766, x ≈ 5.73.

Step 1: Identify the coefficients

For the quadratic equation of the form ax^2 + bx + c = 0, identify the coefficients a, b, and c. Here, a = 2, b = -15, and c = 9.

Step 2: Set up the quadratic formula

The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a). Substitute a, b, and c into the formula with the values identified in Step 1.

Step 3: Calculate the discriminant

Find the value of the discriminant, which is b^2 - 4ac. In this case, it is (-15)^2 - 4(2)(9).

Step 4: Compute the solutions

Calculate the two possible values for x using the plus and minus signs in the formula separately. Then simplify to get the values in decimal form.

Step 5: Check by calculator

Verify the solutions obtained by using a calculator, making sure the answers are rounded to three significant digits.

Key Concepts

Solving Quadratic Equations

Quadratic equations are of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \) and \( c \) are coefficients and \( x \) represents the variable. Solving these equations often leads to finding the value(s) of \( x \) that make the equation true. The most universal method for solving these is the quadratic formula, which is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

To apply this formula, follow these steps:

• Identify the coefficients in the equation. Coefficients are the numerical parts before the variables.
• Substitute these coefficients into the quadratic formula.
• Calculate the discriminant (the part under the square root sign in the formula), as it determines the nature and number of solutions.
• Solve for \( x \) using the plus and minus signs (\pm) in the formula, yielding two possible solutions for \( x \).
• Round off the values to the required number of significant digits for accuracy.

For example, in the equation \( 2x^2 - 15x + 9 = 0 \), the coefficients are \( a = 2 \), \( b = -15 \), and \( c = 9 \). After substituting these values into the quadratic formula, you would complete the operations to find the roots of the equation. This method ensures you can solve any quadratic equation, regardless of its coefficients.

Quadratic Equation Coefficients

In any quadratic equation of the form \( ax^2 + bx + c = 0 \) the coefficients are crucial as they define the curve's characteristics. The coefficient \( a \) affects the width and direction of the parabola, \( b \) impacts the position of the vertex horizontally, and \( c \) determines the height at which the parabola crosses the y-axis.

When approaching a problem:

• Identify each coefficient carefully and assess its sign (+ or –), as the sign can affect the direction of operations in the quadratic formula.
• Remember that the quadratic equation itself represents a parabola when plotted on a graph, and the roots or solutions to the equation are the points where this parabola crosses the x-axis.
• The coefficient \( a \) must not be zero, as it would no longer form a quadratic equation but a linear one instead.

In the context of the provided exercise, \( a = 2 \) which indicates the parabola opens upwards, \( b = -15 \) which will influence the axis of symmetry of the parabola, and \( c = 9 \) which shows the intersection with the y-axis.

Discriminant Calculation

The discriminant in a quadratic equation is a profound concept as it determines the nature and quantity of roots the equation will have. It is denoted as \( b^2 - 4ac \) within the quadratic formula. The value of the discriminant can lead to three scenarios:

• If the discriminant is positive, there will be two distinct real solutions.
• If the discriminant is zero, there will be exactly one real solution (also known as a repeated or double root).
• If the discriminant is negative, there will be no real solutions. Instead, there will be two complex solutions.

To compute the discriminant, square the coefficient \( b \) and then subtract four times the product of coefficients \( a \) and \( c \) from it. In our example, calculating \( (-15)^2 - 4(2)(9) \) gives us 225 - 72, equating to a positive discriminant of 153. Hence, we expect two real solutions for this particular quadratic equation.