Math Problem Statement

let p,q,r be real numbers if px^2 + qx + r = 0 has two real roots α and β where α < -1 and β> 1 find value of 1 + r/p + |q/p|

Solution

We are given the quadratic equation:

px2+qx+r=0p x^2 + q x + r = 0

with two real roots α\alpha and β\beta, where α<1\alpha < -1 and β>1\beta > 1. We are asked to find the value of 1+rp+qp1 + \frac{r}{p} + \left| \frac{q}{p} \right|.

Step 1: Use Vieta's formulas

For a quadratic equation px2+qx+r=0p x^2 + q x + r = 0, Vieta's formulas tell us:

  1. The sum of the roots: α+β=qp\alpha + \beta = -\frac{q}{p}
  2. The product of the roots: αβ=rp\alpha \beta = \frac{r}{p}

Step 2: Use root information

Given that α<1\alpha < -1 and β>1\beta > 1, let's take specific values to simplify the calculations. Let’s choose α=2\alpha = -2 and β=2\beta = 2, which satisfy the condition α<1\alpha < -1 and β>1\beta > 1.

Using these values: α+β=2+2=0qp=0q=0\alpha + \beta = -2 + 2 = 0 \quad \Rightarrow \quad -\frac{q}{p} = 0 \quad \Rightarrow q = 0 αβ=(2)(2)=4rp=4r=4p\alpha \beta = (-2)(2) = -4 \quad \Rightarrow \quad \frac{r}{p} = -4 \quad \Rightarrow r = -4p

Step 3: Calculate the required expression

We need to find the value of: 1+rp+qp1 + \frac{r}{p} + \left| \frac{q}{p} \right| Substitute rp=4\frac{r}{p} = -4 and qp=0\frac{q}{p} = 0: 1+(4)+0=14+0=31 + (-4) + \left| 0 \right| = 1 - 4 + 0 = -3

Final Answer:

The value of 1+rp+qp1 + \frac{r}{p} + \left| \frac{q}{p} \right| is 3-3.

Would you like a deeper explanation or have any questions?


Here are 5 related questions to expand on this problem:

  1. How does Vieta's formulas help solve quadratic equations with real roots?
  2. What conditions guarantee that a quadratic equation has real roots?
  3. How do transformations affect the graph of a quadratic function?
  4. What happens when both roots of a quadratic equation are negative?
  5. How can we factor a quadratic equation with given roots?

Tip: When solving quadratic equations, always check if the discriminant Δ=b24ac\Delta = b^2 - 4ac is non-negative for real roots.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Vieta's Formulas

Formulas

Sum of roots: α + β = -q/p
Product of roots: αβ = r/p
Quadratic equation: px^2 + qx + r = 0

Theorems

Vieta's Formulas

Suitable Grade Level

Grades 9-11