Question

One of the quadratic equation whose roots are A.M. and G.M. of the roots of the equation x² - 10x + 16 = 0, is


x² - 7x + 10= 0
x² - 8x + 15 = 0
x² - 9x + 20= 0
x² - 7x + 12 = 0​

Answer 1

Step-by-step explanation:

Let root of equation

${x}^{2} - 10x + 16$

be p and q . Now

$p + q = 10 \: \: and \: \: pq = 16$

AM of roots will be

$\frac{p + q}{2} = \frac{10}{2} = 5$

and GM of roots

$= \sqrt{pq} = \sqrt{16} = 4$

So equation whose root are 4 and 5 will be

$(x - 4)(x - 5) = 0 \\ \implies {x}^{2} - 9x + 20 = 0$

Or you can also find this equation using

${x}^{2} - ( \alpha + \beta )x + \alpha \beta \\ = {x}^{2} - (4 +5 )x + 4 \times 5 \\ = {x}^{2} - 9x + 20$

Answer 2

Solution :-

comparing given quadratic equation x² - 10x + 16 = 0 with ax² + bx + c = 0 we get,

• a = 1
• b = (-10)
• c = 16

Let roots of the given quadratic equation are p and q .

So,

→ Sum of roots = (-b/a)

→ p + q = -(-10)/1 = 10

then,

→ AM of roots = (p + q)/2 = 10/2 = 5

and,

→ Product of roots = (c/a)

→ p * q = (16/1) = 16

then,

→ GM of roots = √(pq) = √16 = 4

therefore , required quadratic equation :-

→ x² - (sum of roots)x + product of roots = 0

→ x² - (5 + 4)x + 5 * 4 = 0

→ x² - 9x + 20 = 0

Hence, Option (c) x² - 9x + 20 = 0 is correct answer .

Learn more :-

solution of x minus Y is equal to 1 and 2 X + Y is equal to 8 by cross multiplication method

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