Problem 119 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use any convenient method to solve the quadratic equation. $$3 x^{2}-11 x-20=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(3x^{2} - 11x - 20 = 0\) are \(x = 5\) and \(x = -\frac{4}{3}\).

Step 1 - Identify the coefficients

First, identify the coefficients a, b, and c in the equation. In this equation, \(a = 3\), \(b = -11\), and \(c = -20\).

Step 2 - Substitute the values into the quadratic formula

Substitute the coefficients into the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), yielding: \(x = \frac{-(-11) \pm \sqrt{(-11)^{2} - 4 * 3 * -20}}{2 * 3}\). Simplify this to \(x = \frac{11 \pm \sqrt{121 + 240}}{6}\).

Step 3 - Solve to get the solutions

Next, solve the above to get the roots of the equation. This simplifies to \(x = \frac{11 \pm \sqrt{361}}{6}\), which further simplifies to \(x = \frac{11 \pm 19}{6}\). The solutions are therefore \(x = \frac{30}{6} = 5\) and \(x = \frac{-8}{6} = -\frac{4}{3}\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations, which are polynomials that can be written in the standard form of \( ax^2 + bx + c = 0 \). When we talk about solving a quadratic equation, we usually mean finding the values of \( x \) that make the equation true, which are called the roots or solutions of the equation.

The quadratic formula is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This works for any quadratic equation, no matter how complex, because it directly uses the coefficients \(a\), \(b\), and \(c\) from the standard form.

Using the quadratic formula involves three steps: identifying the coefficients from the equation, substituting these values into the formula, and then simplifying the expression to find two possible values for \(x\), which are the roots. These roots might be real or complex numbers, depending on the discriminant \(b^2 - 4ac\).

Roots of Quadratic Equation

In the context of quadratic equations, the term 'roots' refers to the solutions of the equation. These are the values of \(x\) that satisfy the equation \( ax^2 + bx + c = 0 \). The roots can be thought of as the points where the parabola, which is the graph of a quadratic function, intersects the \(x\)-axis.

In solving quadratic equations, we distinguish between different types of roots:

• Real and distinct roots occur when the discriminant \( b^2 - 4ac > 0 \).
• A repeated real root happens when the discriminant is zero \( b^2 - 4ac = 0 \).
• Complex roots come in pairs when the discriminant is negative \( b^2 - 4ac < 0 \).

Finding the roots is critical in various fields such as physics, engineering, and finance because they often represent key points of interest like maximum height of a projectile or break-even points for a business.

Factoring Quadratics

Factoring quadratics is another method of solving quadratic equations, particularly when the quadratic can be expressed as a product of two binomials. This approach is most useful when the equation is factorable over the integers and usually more convenient for simpler, more straightforward quadratics.

Factoring involves rewriting the quadratic equation as \((x - r)(x - s) = 0\), where \(r\) and \(s\) are the roots. If we can write our equation in this factored form, the solutions can be easily found by setting each factor equal to zero and solving for \(x\).

The process of factoring requires finding two numbers that not only multiply to give the constant term, in this case, \(c\), but also add up to the middle coefficient, \(b\). That is, we need two numbers \(p\) and \(q\) such that \(pq = ac\) and \(p + q = b\). If such numbers exist, the quadratic is factorable; if not, we would need to resort to the Quadratic Formula or completing the square to find the roots.