Problem 24 - 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. $$2 x^{2}+5 x-2=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{-5 \pm \sqrt{41}}{4}\).

Step 1: Identify the Coefficients

The given quadratic equation is \(2x^2 + 5x - 2 = 0\). Identify the coefficients \(a\), \(b\), and \(c\) where \(a = 2\), \(b = 5\), and \(c = -2\).

Step 2: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values of \(a\), \(b\), and \(c\) into the formula.\[x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2}\]

Step 3: Calculate the Discriminant

Evaluate the discriminant, \(b^2 - 4ac\):\[5^2 - 4 \cdot 2 \cdot (-2) = 25 + 16 = 41\].

Step 4: Simplify the Expression

Substitute the discriminant back into the quadratic formula:\[x = \frac{-5 \pm \sqrt{41}}{4}\].These are the solutions for the equation.

Step 5: Check Solutions with Sum and Product

Verify the solutions using the sum and product of roots. The sum of the roots \(x_1 + x_2\) should equal \(-\frac{b}{a}\):\[-\frac{5}{2} = \frac{-b}{a} = -2.5\].The product \(x_1 \times x_2\) should equal \(\frac{c}{a}\):\[\frac{-2}{2} = -1\].Both conditions match, indicating the solutions are correct.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The quadratic equation is fundamental in algebra because its solutions are the points where the graph of the equation, a parabola, intersects the x-axis.
Quadratic equations can have:

• Two real and distinct solutions
• One real solution (called a double root)
• Two complex solutions

To solve these equations, the quadratic formula offers a reliable method. Using the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] allows us to find the solutions by calculating the roots directly from the coefficients \(a\), \(b\), and \(c\). This formula considers all possible scenarios of root types, based on the value derived from the discriminant.

Discriminant

The discriminant of a quadratic equation gives insight into the nature of the roots of the equation without fully solving it. It is found using the expression \(b^2 - 4ac\), which is part of the quadratic formula.
Here's what the discriminant can tell us about the roots:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root (or a double root), meaning the parabola touches the x-axis at a single point.
• If the discriminant is negative, the equation has two complex roots, and the parabola does not intersect the x-axis.

In the example given, the discriminant calculation \(5^2 - 4 \cdot 2 \cdot (-2)\) resulted in \(41\), which is a positive value, indicating two distinct real roots for the equation \(2x^2 + 5x - 2 = 0\).

Sum and Product of Roots

The sum and product of the roots of a quadratic equation are closely tied to the equation's coefficients. These relationships provide additional methods to verify solutions.

• The sum of the roots \(x_1 + x_2\) of the quadratic equation \(ax^2 + bx + c = 0\) is given by \( -\frac{b}{a} \).
• The product of the roots \(x_1 \cdot x_2\) is \( \frac{c}{a} \).

These expressions are derived from Vieta's formulas, which are consequences of the structure of quadratic equations.
In checking the equation given, the sum \(-\frac{5}{2}\) and the product \(-1\) calculated from the solutions matched the relationships derived from the coefficients, confirming the solutions were correctly found. This technique is especially useful as a check after using the quadratic formula.