If the roots of the equation x^2 - 7x + p = 0 are in the ratio 3:4, what is the

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Question: If the roots of the equation x^2 - 7x + p = 0 are in the ratio 3:4, what is the value of p?

Options:

Correct Answer: 20

Solution:

Let the roots be 3k and 4k. Then, 3k + 4k = 7 => 7k = 7 => k = 1. The product of the roots is 3k * 4k = 12k^2 = p => p = 12.

If the roots of the equation x^2 - 7x + p = 0 are in the ratio 3:4, what is the value of p?

If the roots of the equation x^2 - 7x + p = 0 are in the ratio 3:4, what is the value of p?

Correct Answer: 12

Step 1: Understand that the roots of the equation are in the ratio 3:4. This means we can express the roots as 3k and 4k, where k is a common multiplier.
Step 2: Write down the sum of the roots. According to Vieta's formulas, the sum of the roots (3k + 4k) should equal the coefficient of x (which is -(-7) = 7). So, we have 3k + 4k = 7.
Step 3: Combine the terms on the left side. This gives us 7k = 7.
Step 4: Solve for k by dividing both sides by 7. This gives us k = 1.
Step 5: Now, find the actual roots using k. Substitute k back into the expressions for the roots: 3k = 3(1) = 3 and 4k = 4(1) = 4.
Step 6: Calculate the product of the roots. The product is 3 * 4 = 12.
Step 7: According to the equation x^2 - 7x + p = 0, the product of the roots is equal to p. Therefore, p = 12.

Quadratic Equations – Understanding the properties of roots of quadratic equations, including their sum and product.
Ratios – Applying the concept of ratios to express the roots in terms of a common variable.
Vieta's Formulas – Using Vieta's formulas to relate the coefficients of the polynomial to the sum and product of its roots.