Problem 106 - Algebra and Trigonometry

Question

Relationship Between Roots and Coefficients The Quadratic Formula gives us the roots of a quadratic equation from its coefficients. We can also obtain the coefficients from the roots. For example, find the roots of the equation \(x^{2}-9 x+20=0\) and show that the product of the roots is the constant term 20 and the sum of the roots is 9 , the negative of the coefficient of \(x\) . Show that the same relationship between roots and coefficients holds for the following equations: $$ \begin{array}{l}{x^{2}-2 x-8=0} \\ {x^{2}+4 x+2=0}\end{array} $$ Use the Quadratic Formula to prove that in general, if the equation \(x^{2}+b x+c=0\) has roots \(r_{1}\) and \(r_{2},\) then \(c=r_{1} r_{2}\) and \(b=-\left(r_{1}+r_{2}\right)\)

Step-by-Step Solution

Answer

Roots and coefficients satisfy the relationships: \(c = r_1 r_2\), \(b = -(r_1 + r_2)\).

Step 1: Use Quadratic Formula to Find Roots of the First Equation

For the equation \(x^2 - 2x - 8 = 0\), we identify coefficients as \(a = 1\), \(b = -2\), and \(c = -8\). Applying the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we calculate:- Discriminant: \(b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36\).- Roots: \(x = \frac{-(-2) \pm \sqrt{36}}{2 \times 1} = \frac{2 \pm 6}{2}\).- Therefore, the roots are \(x = 4\) and \(x = -2\).

Step 2: Verify the Relationship for First Equation

The product of the roots \(r_1 \cdot r_2 = 4 \cdot (-2) = -8\) matches the constant term \(c\). The sum of the roots \(r_1 + r_2 = 4 + (-2) = 2\) is the negative of the coefficient of \(x\), which is \(-(-2) = 2\). Thus, the relationships hold for \(x^2 - 2x - 8 = 0\).

Step 3: Use Quadratic Formula to Find Roots of the Second Equation

For the equation \(x^2 + 4x + 2 = 0\), coefficients are \(a = 1\), \(b = 4\), and \(c = 2\). Applying the quadratic formula:- Discriminant: \(b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 2 = 16 - 8 = 8\).- Roots: \(x = \frac{-4 \pm \sqrt{8}}{2 \times 1} = \frac{-4 \pm 2\sqrt{2}}{2}\).- Therefore, the roots are \(x = -2 + \sqrt{2}\) and \(x = -2 - \sqrt{2}\).

Step 4: Verify the Relationship for Second Equation

The product of the roots \(r_1 \cdot r_2 = (-2 + \sqrt{2}) \cdot (-2 - \sqrt{2}) = 4 - 2 = 2\) matches the constant term \(c = 2\). The sum of the roots \(r_1 + r_2 = (-2 + \sqrt{2}) + (-2 - \sqrt{2}) = -4\) is the negative of the coefficient of \(x\), which is \(-4\). Thus, the relationships hold for \(x^2 + 4x + 2 = 0\).

Step 5: General Proof Using Quadratic Formula

For a general quadratic equation \(x^2 + bx + c = 0\), the quadratic formula gives roots:\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\).The roots are \(r_1 = \frac{-b + \sqrt{b^2 - 4c}}{2}\) and \(r_2 = \frac{-b - \sqrt{b^2 - 4c}}{2}\).- Product of roots \(r_1 \cdot r_2 = \left(\frac{-b + \sqrt{b^2 - 4c}}{2}\right) \cdot \left(\frac{-b - \sqrt{b^2 - 4c}}{2}\right) = \frac{(-b)^2 - (b^2 - 4c)}{4} = c\).- Sum of roots \(r_1 + r_2 = \frac{-b + \sqrt{b^2 - 4c} + (-b - \sqrt{b^2 - 4c})}{2} = \frac{-2b}{2} = -b\).This proves that the product of the roots is equal to \(c\) and their sum is equal to \(-b\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations. It is especially useful for finding the roots of any quadratic equation, which is usually in the form \(ax^2 + bx + c = 0\). With the formula, you don't need to factorize the quadratic expression, which can often be tricky. The quadratic formula is represented as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here’s how it works:
- **Identify the coefficients:** Recognize \(a\), \(b\), and \(c\) in the quadratic equation.
- **Apply the formula:** Plug the values of \(a\), \(b\), and \(c\) into the formula.The part under the square root sign, \(b^2 - 4ac\), is called the discriminant, which will be discussed later. By mastering the quadratic formula, you can easily find whether quadratic equations have real solutions and what these solutions are. This formula will always yield two solutions, or roots, which we call \(x_1\) and \(x_2\).

Roots and Coefficients

Roots of a quadratic equation are simply the values of \(x\) that solve the equation, making the expression equal to zero. These roots have an interesting relationship with the coefficients of the equation. There are two roots in a quadratic equation because it is a polynomial of degree 2.

• The roots are \(x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\) and \(x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\).

The coefficients \(b\) and \(c\) are actually connected to these roots:

• The sum of the roots \((r_1 + r_2)\) is equal to \(-b/a\).
• The product of the roots \((r_1 \cdot r_2)\) is equal to \(c/a\).

This understanding helps in simplifying complex problems or verifying your solutions, as seen in the original exercise's exploration of these relationships. Knowing this can aid in solving quadratic equations quickly and with certainty.

Discriminant

The discriminant is a key component of the quadratic formula and is crucial for understanding the nature of the roots. It's represented by the expression \(b^2 - 4ac\). Depending on the value of the discriminant, you can determine the type and number of roots:

• If the discriminant is positive \((b^2 - 4ac > 0)\), the equation has two distinct real roots.
• If the discriminant is zero \((b^2 - 4ac = 0)\), the equation has exactly one real root, or a repeated root.
• If the discriminant is negative \((b^2 - 4ac < 0)\), the equation has two complex roots.

Understanding the discriminant allows you to anticipate the nature of the solutions without fully solving the equation. This is immensely helpful in situations where you need a quick assessment of the equation's potential solutions, as it indicates whether the solutions are real numbers or complex numbers.

Sum and Product of Roots

The relationships between the roots and coefficients of a quadratic equation are beautifully simple. From the quadratic equation \(x^2 + bx + c = 0\), we find that:

• The sum of the roots \((r_1 + r_2)\) is \(-b/a\).
• The product of the roots \((r_1 \cdot r_2)\) is \(c/a\).

These equations arise from expanding \((x - r_1)(x - r_2) = x^2 - (r_1 + r_2)x + r_1r_2\) and comparing it to \(x^2 + bx + c\).
In practical terms, knowing the sum and product of the roots helps in understanding the equation's properties even before finding the explicit root values. Moreover, these relationships are handy for verifying solutions. In simpler cases, they can even provide the roots directly without much hassle, ensuring you're aware of any errors in your calculations.