Problem 19 - Algebra 2 and Trigonometry

Question

One of the roots is given. Find the other root. \(x^{2}-8 x+c=0 ;-3\)

Step-by-Step Solution

Answer

The other root is 11.

Step 1: Understand the Problem

We are given a quadratic equation \(x^{2} - 8x + c = 0\), and one of the roots is \(-3\). We need to find the other root of this equation.

Step 2: Use the Root to Find c

Since \(-3\) is a root of the equation, substitute \(x = -3\) into the equation to find \(c\). This gives: \((-3)^{2} - 8(-3) + c = 0\), simplifying to \(9 + 24 + c = 0\). Thus, \(c = -33\).

Step 3: Write the Complete Equation

With \(c = -33\), the equation becomes \(x^{2} - 8x - 33 = 0\).

Step 4: Use Vieta's Formulas

According to Vieta's formulas, the sum of the roots \(r_1 + r_2\) of a quadratic equation \(x^2 + bx + c = 0\) is \(-b\) (which is 8 in this case), and the product \(r_1 \cdot r_2\) is \(c\) (which is -33). With \(r_1 = -3\), we solve \(-3 + r_2 = 8\), so \(r_2 = 11\).

Step 5: Verify the Solution

Check that the roots \(-3\) and \(11\) satisfy the original equation \(x^2 - 8x - 33 = 0\). The product of the roots is \((-3) \times 11 = -33\), which matches the constant term, and the sum is \(-3 + 11 = 8\), which matches the negative coefficient of \(x\). Thus, both calculations confirm the solution.

Key Concepts

Vieta's Formulas

Understanding Vieta's formulas is a great way to find relationships between the roots and coefficients of quadratic equations.

For any quadratic equation framed as \(x^2 + bx + c = 0\), Vieta's formulas help us determine two things:

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

Applying this to our equation \(x^2 - 8x - 33 = 0\), you see that the sum of the roots must be 8 (from \(-(-8)\)) and their product must be \(-33\). With one root known as \(-3\), you can use these relationships to efficiently find the other root. This is a useful tool for solving many quadratic equation problems quickly.

Finding Roots

Finding the roots of a quadratic equation like \(x^2 - 8x - 33 = 0\) involves determining the values of \(x\) that satisfy the equation.

With one root given, such as \(-3\), use it to identify constants and find the other root by using algebraic methods or Vieta's formulas.

• Start with the sum of the roots formulation \(-3 + r_2 = 8\).
• This means \(r_2 = 11\).

These roots \(-3\) and \(11\) fit perfectly due to their sum \(8\) and product \(-33\). Remember, each root represents a solution where the expression \(x^2 - 8x - 33 = 0\) equals zero.

Substituting Values

Substituting known values allows us to solve for unknowns or verify solutions in a quadratic equation.

When given that \(-3\) is a root of \(x^2 - 8x + c = 0\), plug \(x = -3\) into the equation:

• \((-3)^2 - 8(-3) + c = 0\).
• Simplifying yields \(9 + 24 + c = 0\), so \(c = -33\).

Substituting values is crucial for manipulating equations to find other missing parts. Once \(c\) is found, the equation turns into \(x^2 - 8x - 33 = 0\). By substituting \(-3\) and \(11\) back into this modified equation, you can confirm they satisfy it, affirming these values as the correct roots.