A ball is thrown up from the top of an 8 m tall building. Its height after t seconds is h = 8 + 2t − t², drawn in the graph. When does the ball hit the ground?

t = 4 s (the other root, t = −2, is rejected). The ball is on the ground when h = 0 : -t^2 + 2t + 8 = 0 t^2 - 2t - 8 = 0 (t - 4)(t + 2) = 0 . The graph meets the t-axis at t = 4 ; time cannot be negative, so t = -2 is rejected.

A ball is thrown up from the top of an 8 m tall building. Its height after $t$ seconds is $h = 8 + 2 t - t^{2}$ , drawn in the graph. When does the ball hit the ground?

Class 10Quadratic EquationsAlgebra — Quadratics

At = 1 s, when the ball is highest
Bt = 4 s (the other root, t = −2, is rejected)
Ct = 2 s (the other root, t = −4, is rejected)
Dt = 9 s, from the highest point of the graph

Step-by-Step Solution

The ball is on the ground when

h = 0 : - t^{2} + 2 t + 8 = 0 \Rightarrow t^{2} - 2 t - 8 = 0 \Rightarrow (t - 4) (t + 2) = 0

. The graph meets the t-axis at

t = 4

; time cannot be negative, so

t = - 2

is rejected.

Answer: B) t = 4 s (the other root, t = −2, is rejected)

Key idea: The discriminant — nature of roots

D = b^{2} - 4 a c

classifies the roots of

a x^{2} + b x + c = 0 : - * * D > 0

**: TWO distinct real roots. Parabola crosses

x

-axis at two points.
- **

D = 0

**: ONE repeated real root (called a 'double root' or 'two equal roots'). Parabola touches

x

-axis at the vertex.
- **

D < 0

**: NO real roots. Parabola does not touch the

x

-axis at all.

**Use case**: 'For what values of

k

does

k x^{2} + 2 x + 1 = 0

have equal roots?' → set

D = 0 : 4 - 4 k = 0

→

k = 1

.

You can answer 'how many real roots?' WITHOUT computing them — just compute

D

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A ball is thrown up from the top of an 8 m tall building. Its height after t seconds is h=8+2t−t2, drawn in the graph. When does the ball hit the ground?

Step-by-Step Solution

Key idea: The discriminant — nature of roots

Master this chapter in 30 minutes a day.

More Class 10 Quadratic Equations practice

A ball is thrown up from the top of an 8 m tall building. Its height after $t$ seconds is $h = 8 + 2 t - t^{2}$ , drawn in the graph. When does the ball hit the ground?