Math Problem Statement

the equation x power 2 +px + q = 0 where p and q are constant , has roots -3 and 5 so find the values of p and q

Solution

Given the quadratic equation:

x2+px+q=0x^2 + px + q = 0

where pp and qq are constants, and the roots are 3-3 and 55, we can use the relationships between the roots and coefficients of a quadratic equation.

For a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, with roots α\alpha and β\beta, the relationships are:

  1. Sum of the roots: α+β=ba\alpha + \beta = -\frac{b}{a}
  2. Product of the roots: αβ=ca\alpha \cdot \beta = \frac{c}{a}

In your case:

  • α=3\alpha = -3 and β=5\beta = 5
  • The equation is in the form x2+px+q=0x^2 + px + q = 0, so a=1a = 1, b=pb = p, and c=qc = q.

Step 1: Find pp

Using the sum of the roots:

α+β=p\alpha + \beta = -p

Substitute α=3\alpha = -3 and β=5\beta = 5:

3+5=p-3 + 5 = -p

This simplifies to:

2=p2 = -p

Thus, p=2p = -2.

Step 2: Find qq

Using the product of the roots:

αβ=q\alpha \cdot \beta = q

Substitute α=3\alpha = -3 and β=5\beta = 5:

35=q-3 \cdot 5 = q

This simplifies to:

q=15q = -15

Final answer:

  • p=2p = -2
  • q=15q = -15

Would you like more details on these steps?


Here are 5 related questions to expand your understanding:

  1. How can you verify the roots of a quadratic equation once you have found pp and qq?
  2. What is the general formula for solving a quadratic equation?
  3. How does the discriminant of a quadratic equation help in determining the nature of its roots?
  4. What happens to the sum and product of roots if the coefficients of the quadratic equation change?
  5. How would the equation change if the roots were complex?

Tip: Always check your final result by plugging the values of pp and qq back into the original equation to ensure it holds true for the given roots.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Sum and Product of Roots

Formulas

Sum of roots: α + β = -b/a
Product of roots: α * β = c/a
Quadratic equation: ax^2 + bx + c = 0

Theorems

Relationships between roots and coefficients

Suitable Grade Level

Grades 9-10