Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find two quadratic equations having the given solutions. (There are many correct answers.) $$5 \sqrt{3},-5 \sqrt{3}$$

Step-by-Step Solution

Answer

Two possible quadratic equations having the roots \(5 \sqrt{3}\) and \(-5 \sqrt{3}\) are \(ax^2 - 75a = 0\) and \(bx^2 - 75b = 0\), where \(a\) and \(b\) are any real numbers.

Step 1: Substitute the roots into the equation

Let the roots of the quadratic equations be \(5 \sqrt{3}\) and \(-5 \sqrt{3}\). Thus, two quadratic equations can be written as follows:\n\(a(x - 5 \sqrt{3})(x + 5 \sqrt{3}) = 0\)and\(b(x - 5 \sqrt{3})(x + 5 \sqrt{3}) = 0\),where \(a\) and \(b\) are any real numbers.

Step 2: Apply the distributive law and simplify

Apply the distributive law (also known as the FOIL method in this case: First, Outer, Inner, Last) to \( (x - 5 \sqrt{3})(x + 5 \sqrt{3})\):First, \(x * x\) makes \(x^2\)\nOuter, \(x * 5 \sqrt{3}\) makes \(5x \sqrt{3}\)\nInner, \(-5\sqrt{3} * x\) makes \(-5x \sqrt{3}\)\nLast, \(-5 \sqrt{3} * 5 \sqrt{3}\) makes \(-75\)Add these four parts together:\(x^2 - 75 = 0\)Thus, the two quadratic equations are:\(ax^2 - 75a = 0\)and\(bx^2 - 75b = 0\).

Step 3: Write down the final quadratic equations

So, two quadratic equations having the roots \(5 \sqrt{3}\) and \(-5 \sqrt{3}\) are:\(ax^2 - 75a = 0\)and\(bx^2 - 75b = 0\),where \(a\) and \(b\) are any real numbers.

Key Concepts

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of the variable that make the equation true. For a quadratic equation in the form of \(ax^2 + bx + c = 0\), the roots can be real or complex numbers.
If you know the roots in advance, you can form a quadratic equation by working backwards. For example, if the roots are \(r_1\) and \(r_2\), the equation can be expressed as \((x - r_1)(x - r_2) = 0\).
In the example given, the roots of the quadratic equation are \(5 \sqrt{3}\) and \(-5 \sqrt{3}\). Using these roots, one can form various quadratic equations. The general form is \(k(x - 5\sqrt{3})(x + 5\sqrt{3}) = 0\), where \(k\) is any real number.

Distributive Law in Algebra

The distributive law is a vital property in algebra. It explains how to multiply a term by a sum or difference. Specifically, it allows us to spread out the multiplication over addition: \(a(b + c) = ab + ac\).
This law is used explicitly in solving quadratic expressions, especially when dealing with brackets. It's the reason we can expand \((x - 5 \sqrt{3})(x + 5 \sqrt{3})\) into a quadratic form.
More formally, every term in the first bracket is multiplied by every term in the second bracket. This ensures that we account for every combination of original terms. The result here is \(x^2 - 75\). This example uses both the distributive law and the principle of conjugates to simplify into a real number.

FOIL Method

The FOIL method is a specific application of the distributive law used for binomials. It stands for First, Outer, Inner, Last and helps multiply two binomials systematically.
Let's apply FOIL to \((x - 5\sqrt{3})(x + 5\sqrt{3})\):

• **First:** Multiply the first terms of each binomial: \(x \cdot x = x^2\).


• **Outer:** Multiply the outer terms: \(x \cdot 5\sqrt{3} = 5x\sqrt{3}\).


• **Inner:** Multiply the inner terms: \(-5\sqrt{3} \cdot x = -5x\sqrt{3}\).


• **Last:** Multiply the last terms: \(-5\sqrt{3} \cdot 5\sqrt{3} = -75\).

When combined, the middle terms cancel out, resulting in \(x^2 - 75\). This technique is invaluable in algebra to simplify and solve expressions.

Real Numbers in Algebra

In algebra, real numbers include all numbers on the number line. This means all rational and irrational numbers, but no imaginary numbers.
When forming quadratic equations, coefficients can be any real number. Both positive and negative values of real numbers are included. This flexibility allows the substitution of any real number for \(a\) or \(b\) in our derived equations \(ax^2 - 75a = 0\) and \(bx^2 - 75b = 0\).
Using real numbers ensures the equations describe values we can graph and understand intuitively. The real numbers \(5\sqrt{3}\) and \(-5\sqrt{3}\) are not only roots, but also critical in defining the shape and position of the quadratic function on a graph.