Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. \(50 x^{2}-60 x+10=0\)

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x_{1} = 1\) and \(x_{2} = 0.2\).

Step 1: Identify coefficients

Identify the coefficients for the quadratic equation, which are \(a = 50\), \(b = -60\), and \(c = 10\)

Step 2: Calculate the discriminant

Calculate the discriminant value using the formula \(D = b^{2} - 4ac\). Substituting the values \(a = 50\), \(b = -60\), and \(c = 10\) we get, \(D = (-60)^{2} - 4*50*10 = 3600 - 2000 = 1600\)

Step 3: Apply quadratic formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to solve for \(x\). Substituting the values \(a = 50\), \(b = -60\), and \(D = 1600\), we have two solutions: \(x_{1} = \frac{60 + \sqrt{1600}}{100} = 1\) and \(x_{2} = \frac{60 - \sqrt{1600}}{100} = 0.2\)

Key Concepts

Discriminant

The discriminant is a key component of the quadratic formula and gives us a wealth of information about the solutions of a quadratic equation. In the context of a quadratic equation written as \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). This small but mighty value determines the nature and number of solutions we can expect.For quadratic equations:

• If \(D > 0\), there are two distinct real roots. This means the parabola intersects the x-axis at two different points.
• If \(D = 0\), there is exactly one real root, indicating that the parabola just touches the x-axis, known as a repeated or double root.
• If \(D < 0\), there are no real roots, and instead, the roots are complex or imaginary, meaning the parabola does not intersect the x-axis at all.

In the original exercise, we calculated the discriminant to be 1600, which is positive. This confirmed there are two distinct real solutions for the equation \(50x^2 - 60x + 10 = 0\). Understanding the discriminant helps us predict what lies ahead before we dive into solving the equation.

Quadratic Formula

The quadratic formula is like a magic key that unlocks the roots of any quadratic equation, no matter how complex it seems. This formula is particularly useful when factoring is not straightforward. For any quadratic equation \(ax^2 + bx + c = 0\), the quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula demands only three coefficients — \(a\), \(b\), and \(c\) — and with these, plus the discriminant, it computes the values of \(x\) that solve the equation.Here's a step-by-step rundown:

• Compute the discriminant \(b^2 - 4ac\) (we completed this in the previous step).
• Find \(-b\), which represents the opposite of the coefficient \(b\).
• Calculate \(2a\), doubling the coefficient \(a\).
• Plug these into the formula to solve for \(x\), noting the \(\pm\) sign, which signals two possible roots.

In the given example, substituting \(a = 50\), \(b = -60\), and the discriminant \(1600\) gives us \(x_1 = 1\) and \(x_2 = 0.2\). The beauty of this formula lies in its versatility to handle any quadratic equation with ease.

Coefficients

Coefficients are the numerical or constant part of the terms in a polynomial equation and play a pivotal role in determining the behavior and solutions of the equation. In the standard form of a quadratic equation \(ax^2 + bx + c = 0\), the coefficients \(a\), \(b\), and \(c\) represent different parts:

• \(a\) is the coefficient of \(x^2\), and it influences the "width" and orientation of the parabola (whether it opens up or down).
• \(b\) is the coefficient of \(x\), affecting the parabola's tilt or direction on the graph.
• \(c\) is the constant term and determines where the parabola crosses the y-axis.

In the provided equation \(50x^2 - 60x + 10 = 0\), the coefficients are \(a = 50\), \(b = -60\), and \(c = 10\). Recognizing these coefficients allowed us to correctly apply the quadratic formula and compute the solutions. Each coefficient has a dedicated role in shaping the graph of the equation, making them essential to understand in solving and graphing quadratic equations.