Question

The difference between the roots of the equation
x^2-13x+k=0 is 7 find k
Note-need to be solved by step

Answer 1

Answer:

30

Step-by-step explanation:

Let α and β are roots of the equation.

//if quadratic equation is of form ax² + bx + c then sum of roots = -b/a; product of roots = c/a.

quadratic equation is x^2-13x+k=0

Sum of roots = α + β = - b/a = 13

difference of roots = α - β = 7

solve α + β = 13 and α - β = 7

α = 10; β = 3.

product of roots = αβ = c/a = k

=> k = 10*3 = 30

Answer 2

Answer:

The value of k is

$\boxed{\red{\sf\:k\:=\:30}}$

Step-by-step-explanation:

The given quadratic equation is x² - 13x + k = 0.

Comparing with ax² + bx + c = 0, we get,

• a = 1,

• b = - 13

• c = k

Let the roots of the given quadratic equation be $\sf\:\alpha\:\&\:\beta$

Now, we know that,

$\sf\:Sum\:of\:roots\:=\:\alpha\:+\:\beta\:=\:-\:\frac{b}{a}\\\\\implies\sf\:\alpha\:+\:\beta\:=\:-\:(\:-\:\frac{13}{1}\:)\\\\\implies\sf\:\alpha\:+\:\beta\:=\:13\\\\\sf\:Now,\\\\\sf\:Product\:of\:roots\:=\:\alpha.\:\beta\:=\:\frac{c}{a}\\\\\implies\sf\:\alpha.\:\beta\:=\:\frac{k}{1}\\\\\implies\sf\:\alpha.\:\beta\:=\:k$

Now, from the given condition,

The difference between the roots of quadratic equation is 7.

$\therefore\sf\:\alpha\:-\:\cancel{\beta}\:=\:7\\\\\sf\:\alpha\:+\:\cancel{\beta}\:=\:13\\\\\implies\sf\:2\:\alpha\:=\:20\:\:\:[\:Adding\:both\:equations\:]\\\\\implies\sf\:\alpha\:=\:\cancel{\frac{20}{2}}\\\\\implies\sf\:\alpha\:=\:10$

Now,

$\sf\:\alpha.\:\beta\:=\:k\\\\\implies\sf\:10\:\times\:3\:=\:k\\\\\implies\boxed{\red{\sf\:k\:=\:30}}$