Question

obtain the quadratic equation if roots are -3,-7​

Answer 1

Answer:

The quadratic equation with roots -3 and -7 is x²+10x+21=0

Step-by-step explanation:

Given: Roots say,

α=-3 and β=-7

To find: Quadratic equation with the two roots given above.

Solution:

We know the product of two roots α.β (For any equation ax²+bx+c)

Sum of roots = α+β (For any equation ax²+bx+c)

Therefore,

The Product of roots here is = α.β= (-3).(-7)=21 ----------------(1)

The Sum of the roots = α+β= (-3)+(-7)=-10---------------------(2)

Now solving equations 1 and 2 to find α and β

By applying the property:

(a+b)² = a²+b²+2ab,

(α+β)²=α²+β²+2αβ

α+β is -10 and αβ = 21

(-10)²= α²+β²+(2) (21)

100 = α²+β²+42

α²+β²=100-42 = 58-----------------(3)

Applying another property and using equation 3,

(α-β)²=α²+β²-2αβ,

(α-β)²=58 - 2.21= 58-42 =16,

α-β=√16 = 4--------------------------(4)

So we have two equations (2) and (4) And we will solve them in order to get the values of α and β.

α+β= -10

α- β= 4

Adding the above equations,

2α = -6

α = -3

Subtracting above equations,

2β=-14

β=-7

So equation is written as:

(x-α)(x-β)=0

(x+3)(x+7)=0

Therefore the answer is  : x²+10x+21=0