Quadratic Equations

We are given a quadratic in with parameter , and are asked to find how many real values of make the equation have distinct roots. Let us write the general quadratic: Compute the discriminant: Where:
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- So: We want: This is a quadratic inequality. Since the parabola opens upward (coefficient of is positive), the expression is positive outside the roots.
Solve:



So, Therefore, the quadratic has distinct roots for: This is an infinite number of real values of , hence:

We are given two quadratic equations: We are told that these two equations have one common root, say .
Step 1:
Let the common root of both equations be . Substituting into both equations: From Equation 1: From Equation 2:
Step 2:
Now, subtract Equation (1) from Equation (2) to eliminate the constant terms: Simplifying: Factoring:
Step 3:
For this equation to hold, either or . - If , substitute into Equation (1): Therefore, is not a valid root. - Thus, we must have .
Step 4:
Substitute into Equation (1): Expanding: Now, solve this quadratic equation for using the quadratic formula: Thus, the two possible values for are:
Step 5:
The sum of all possible values of is: Thus, the correct answer is:

Step 1: Use the identity for the sum of reciprocals of roots.
We are given the quadratic equation:
3x² - 7x + 2 = 0
Let the roots be x₁ and x₂. We need to calculate:
1/x₁ + 1/x₂ = (x₁ + x₂) / (x₁ × x₂)

Step 2: Apply Vieta’s formulas.
For any quadratic equation ax² + bx + c = 0:

• Sum of roots x₁ + x₂ = -b/a = -(-7)/3 = 7/3
• Product of roots x₁ × x₂ = c/a = 2/3

Step 3: Substitute the values into the formula.
(x₁ + x₂) / (x₁ × x₂) = (7/3) / (2/3) = (7/3) × (3/2) = 7/2

Final Answer: 7/2

Step 1: Use the conditions for real and distinct roots.
For a quadratic equation , the roots are real and distinct if the discriminant .
Given the equation , here , , and .
So, the discriminant is:
Since the roots are real and distinct, we must have: Divide by 4: This implies: Step 2: Use the condition that the roots differ utmost by .
Let the roots of the equation be and .
From Vieta's formulas for :
Sum of roots:
Product of roots:
The difference between the roots is . We know the identity: Substitute the sum and product of roots: So, the absolute difference between the roots is: We are given that the roots differ utmost by , which means .
Divide by 2: Since both sides are non-negative (because from Condition 1 and is a positive quantity), we can square both sides without changing the inequality direction: Rearrange the inequality to solve for : Multiply by -1 and reverse the inequality sign: Step 3: Combine the conditions for .
From Condition 1, we have .
From Condition 2, we have .
Combining these two inequalities, we find the interval for :
Thus, lies in the interval .

Given: The quadratic equation has sum of roots and product of roots . For the equation: we identify the coefficients as:
Step 1: Find the sum of the roots This matches the given sum of the roots.
Step 2: Find the product of the roots
Step 3: Use the product and test values of Since the problem does not directly specify the product, we check possible values of that yield roots with sum 5.
Step 4: Check discriminant for real roots For real roots, , but this is not strictly required by the problem.
Step 5: Test Factoring: Roots are and . Check sum and product: Sum matches the given value and product equals .
Therefore, the value of is: