For any quadratic equation ax² + bx + c = 0 with roots α and β,

Sum of roots , α + β = -b/a
Product of roots, αβ = c/a

Since you already know k = 7, then y = x² - 10x - 7

As the graph cuts the x-axis (where y = 0) at (α,0) and (β,0), the roots of the equation x² - 10x - 7 are α and β.

These are the intersections of the function y = x² - 10x - 7 and line y = 0 (effectively the x-axis), where their values equal each other

i.e x² - 10x - 7 = 0

So for this function, a = 1, b = -10, c = -7

Sum of roots, α + β = -(-10)/1 = 10
Product of roots, αβ = -7/1 = -7


Next, α³ + β³ can be found by using the property (α + β)³ = α³ + 3α²β + 3αβ² + β²

(You can choose to expand the left hand side term to verify it)


Then,

(α + β)³ = α³ + 3αβ(α + β) + β³

α³ + β³ = (α + β)³ - 3αβ(α + β)

= 10³ - 3(-7)(10)

= 1000 + 210

= 1210

Alternatively, use the property

α³ + β³ = (α + β)(α² - αβ + β²)

(This is basically a rewritten form of the previous property)

Then, rewrite this as :

(α + β)(a² + 2αβ + β² - 3αβ)

= (α + β)((α + β)² - 3αβ)

(Recall the property a² + 2ab + b² = (a + b)²)

= (10)(10² - 3(-7))

= 10(100 + 21)

= 10(121)

= 1210

In case your unsure why the formula for sum and product of roots are like this, consider the following :


When y = 0, (x - α)(x - β) = 0

x² - βx - αx - α(-β) = 0

x² - (α + β)x + αβ = 0 ①

Compare this to :

ax² + bx + c = 0
x² + b/a x + c/a = 0
x² - (-b/a) x + c/a = 0 ②


From comparing ① and ②, we deduce that sum of roots = -b/a and product of roots = c/a