Question

Find the value of p such that the quadratic equation x^2-
(2p+1)x+(3p +7) =0 has sum of the roots as one-third of their product.​

Answer 1

Answer:

The value of p is $\frac{-10}{9}$

Explanation:

The quadratic equation is ax²+bx+c = 0, where 'x' is a variable and 'a', 'b', and 'c' are coefficients.

The sum of the roots is equal to the negative of the second term's coefficient, divided by the leading coefficient. The third term divided by the first term represents the product of roots.

Here,

a = 1

b = 2p+1

c = 3p +7

From the question, we have

sum of the roots is equal to one-third of their product.

$\frac{-b}{a} = \frac{1}{3} *\frac{c}{a} \\\frac{-(2p+1)}{1} = \frac{1}{3} *\frac{3p +7}{1} \\3p+7 = -3*(2p+1)\\3p+7 = -6p-3\\9p = -10\\p = \frac{-10}{9}$

The value of p is $\frac{-10}{9}$

Multiplication:

Multiplying the numbers to find the sum of two or more numbers. Multiplication is the basic mathematical method that is widely used in daily life. The multiplicand, multiplier, and product are the elements of it. The first number is the multiplicand, the second number is the multiplier, and the result is the product.  The results of multiplying two or more numbers is referred to as the product of those numbers, and the factors are the quantities that are multiplied.

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Answer 2

Answer:

The value of p that satisfies the given condition is 1.

Step-by-step explanation:

Let the roots of the quadratic equation x^2 - (2p+1)x + (3p+7) = 0 be denoted as α and β.

We know that the sum of the roots of a quadratic equation is given by α + β and the product of the roots is given by α × β.

From the given condition, we can write:

α + β = (1/3) α β

Multiplying both sides by 3, we get:

3(α + β) = α β

Expanding the left side and using the fact that α + β is equal to the coefficient of x with opposite sign, we get:

3(α + β) = (2p + 1) (α + β)

Simplifying, we get:

3 = 2p + 1

2p = 2

p = 1

Therefore, the value of p that satisfies the given condition is 1.

To check this, we can substitute p = 1 into the given quadratic equation and find the roots using the quadratic formula. The roots turn out to be α = 2 and β = 1.5, and their sum is 3.5, which is one-third of their product (2 × 1.5 = 3).

Hence, the value of p that satisfies the given condition is 1.

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