Problem 45 - Algebra for College Students

Question

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$12 x^{2}-73 x+110=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{10}{3}\) and \(x = \frac{11}{4}\).

Step 1: Identify the coefficients

In the quadratic equation \(12x^2 - 73x + 110 = 0\), identify the coefficients: \(a = 12\), \(b = -73\), and \(c = 110\).

Step 2: Write the quadratic formula

The quadratic formula is given by \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\).

Step 3: Calculate the discriminant

Compute the discriminant: \(b^2 - 4ac = (-73)^2 - 4 \times 12 \times 110\). So the discriminant is \(5329 - 5280 = 49\).

Step 4: Find the square root of the discriminant

The square root of the discriminant \(49\) is \(7\).

Step 5: Apply the quadratic formula

Substitute the values into the quadratic formula: \(x = \frac{{73 \pm 7}}{24}\).

Step 6: Solve for the roots

Calculate the two roots: \(x_1 = \frac{{73 + 7}}{24} = \frac{80}{24} = \frac{10}{3}\), and \(x_2 = \frac{{73 - 7}}{24} = \frac{66}{24} = \frac{11}{4}\).

Step 7: Verify using sum and product

According to Vieta's formulas, the sum of the roots \(x_1 + x_2 = \frac{10}{3} + \frac{11}{4} = \frac{40}{12} + \frac{33}{12} = \frac{73}{12}\), should equal \(\frac{b}{a} = \frac{73}{12}\), and the product \(x_1 \cdot x_2 = \frac{10}{3} \cdot \frac{11}{4} = \frac{110}{12}\), should equal \(\frac{c}{a} = \frac{110}{12}\). This confirms the solutions.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). These equations result in a parabola when graphed on a coordinate plane.
Quadratic equations are vital in algebra because they describe numerous natural and physical processes.

• Standard Form: The typical equation is expressed as \(ax^2 + bx + c = 0\).
• Quadratic Terms: Includes terms with \(x^2\), \(x\), and the constant term \(c\).
• Key Point: The highest degree term in the equation, the \(x^2\) term, signifies it being quadratic.

Solving a quadratic equation finds the roots, or values of \(x\), that make the equation true. You often use methods like factoring, completing the square, or the quadratic formula.

Discriminant

The discriminant is part of the quadratic formula and tells us about the nature of the roots of a quadratic equation. The discriminant \(D\) is found using the formula \(b^2 - 4ac\).
This value determines whether the roots are real and how many different solutions there are.

• Positive \(D\): The equation has two distinct real roots, as in the example where \(D = 49\).
• Zero \(D\): There is exactly one real root, meaning the parabola touches the x-axis at one point.
• Negative \(D\): Indicates two complex roots, with no real number solutions.

Understanding the discriminant helps in quickly determining the number and types of solutions for quadratic equations.

Vieta's Formulas

Vieta's formulas provide a relationship between the coefficients of a polynomial and its roots. They are especially useful for simplifying the process of verifying solutions.
For a quadratic equation \(ax^2 + bx + c = 0\) with roots \(x_1\) and \(x_2\), Vieta's formulas state:

• Sum of the roots: \(x_1 + x_2 = -\frac{b}{a}\)
• Product of the roots: \(x_1 \cdot x_2 = \frac{c}{a}\)

These formulas came in handy for verifying the solutions to the quadratic equation \(12x^2 - 73x + 110 = 0\). They ensured that the sum and product of \(x_1 = \frac{10}{3}\) and \(x_2 = \frac{11}{4}\) matched the expected results.

Roots of a Polynomial

The roots of a polynomial are the values of \(x\) that make the polynomial equal to zero. For a quadratic equation, these are the solutions we look for.
Finding roots of quadratic equations can be done using various methods, but the quadratic formula is one of the most reliable and straightforward methods.

• Quadratic Roots: Two solutions calculated using the quadratic formula in the form \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Verification: Roots can be checked by substituting back into the original equation or using Vieta's formulas to check consistency.
• Real vs. Complex: The discriminant indicates whether the roots are real numbers or complex.

In the example, the roots \(x_1 = \frac{10}{3}\) and \(x_2 = \frac{11}{4}\) are both real, calculated neatly with the quadratic formula. This illustration underscores the roots' role in solving and understanding the behavior of quadratic equations.