Question

If \(\mathrm{A}\) and \(\mathrm{B}\) are the roots of the quadratic equation \(\mathrm{x}^{2}-12 \mathrm{x}+27=0\), then \(\mathrm{A}^{3}+\mathrm{B}^{3}\) is (1) 27 (2) 729 (3) 756 (4) 64

Short answer

Answer: The sum of the cubes of the roots is 756.

Step-by-step solution

Find the roots A and B

Use the quadratic formula to find the roots of the given quadratic equation \(x^2 - 12x + 27 = 0\). The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) In our case, a = 1, b = -12, and c = 27. Plugging in these values, we get: \(x = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 27}}{2 \cdot 1}\)

Simplify the expression for the roots

Simplify the expression for the roots by solving the equation under the square root and performing the calculations: \(x = \frac{12 \pm \sqrt{144 - 108}}{2}\) \(x = \frac{12 \pm \sqrt{36}}{2}\) \(x = \frac{12 \pm 6}{2}\)

Calculate the roots A and B

Now, calculate the two roots using the two possible values of the plus-minus sign: A = \(\frac{12 + 6}{2} = 9\) B = \(\frac{12 - 6}{2} = 3\)

Calculate A^3 and B^3

Next, cube the values of A and B: \(A^3 = 9^3 = 729\) \(B^3 = 3^3 = 27\)

Calculate the sum A^3 + B^3

Add the cubes of the roots A and B: \(A^3 + B^3 = 729 + 27 = 756\) Hence, the required sum of the cubes of the roots is 756. Therefore, the correct answer is (3) 756.

Key concepts

Roots of Quadratic Equation

In algebra, a quadratic equation is an expression of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The solutions to this equation are known as the roots. These roots can be real or complex numbers based on the discriminant value. To find these roots, we often use the **quadratic formula**: \[ \text{x} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula helps us calculate the two roots, known commonly as \(A\) and \(B\) in the context of this exercise. When finding these values, remember to:

• Identify \(a\), \(b\), and \(c\) from your equation.
• Calculate the discriminant \(b^2 - 4ac\).
• Use the quadratic formula, applying correct algebraic techniques like completing the square, where needed.

For example, given the quadratic equation \(x^2 - 12x + 27 = 0\), we calculate the roots \(A = 9\) and \(B = 3\) using the formula. Watch out for plus-minus signs as they yield two distinct solutions.

Cubic Functions

A cubic function is a polynomial of degree three, represented in the form \(f(x) = ax^3 + bx^2 + cx + d\). In the context of quadratic roots, one intriguing problem is finding the cubes of these roots. This procedure involves calculating \(A^3\) and \(B^3\) if \(A\) and \(B\) are the roots of a quadratic function. In our solved problem, we find \(A = 9\) and \(B = 3\) using the quadratic equation. Therefore, to obtain the cubic powers, we compute:

• \(A^3 = 9^3 = 729\)
• \(B^3 = 3^3 = 27\)

Understanding cubic functions grants deeper insights into polynomial behavior and enhances algebraic manipulation skills. They often serve as crucial steps in solving complex equations and mathematical models across different applications.

Sum of Cubes of Roots

Once the cubes of the roots are known, finding their sum is a straightforward arithmetic exercise. If \(A\) and \(B\) are the roots of a quadratic and their cubes are \(A^3\) and \(B^3\), the sum \(A^3 + B^3\) becomes crucial in various mathematical contexts. For our given roots \(A = 9\) and \(B = 3\), we calculated their cubes as follows:

• \(A^3 = 729\)
• \(B^3 = 27\)

Hence, by adding these values, the sum \(A^3 + B^3 = 729 + 27 = 756\). This calculation demonstrates the application of algebraic functions and simulates more complex algebraic operations useful in higher-level mathematics and problem-solving.