Page 63 - Math Course 3 (Book 1)

The Quadratic Formula

SPACE TRAVEL Since a negative number of seconds is not
reasonable, use the positive solutions.
The height H of an object t seconds after it is

propelled upward with an initial velocity v is
1 Answer
2
represented by H = – gt + vt + h, where g is the
2
gravitational pull and h is the initial height. A ball thrown on Mars will stay aloft 5.6 – 2.2
or about 3.4 seconds longer than the ball
In order to find when the ball hits the ground, you thrown on Earth. The ball thrown on Europa
must find when H = 0. Write two equations to will stay aloft 15.6 – 2.2 or about 13.4 seconds
represent the situation on Mars and on Europa. longer than the ball thrown on Earth.

Baseball Thrown on Mars Use the Discriminant

1
H = – gt + vt + h
2
2 Example
1
2
0 = – (3.7)t + 10t + 2
2
State the value of the discriminant for
2
2
0 = –1.85t + 10t + 2 3x + 10x = 12. Then determine the number
of real roots of the equation.
Baseball Thrown on Europa Step 1 Rewrite the equation in standard form.
1
2
2
H = – gt + vt + h 3x + 10x = 12 Original equation
2
1
2
2
0 = – (1.3)t + 10t + 2 3x + 10x – 12 = 12 – 12 Subtract 12 from
2 each side.
2
0 = –0.65t + 10t + 2 3x + 10x – 12 = 0 Simplify.
2
These equations cannot be factored, and
completing the square would involve a lot Step 2 Find the discriminant.
of computation.
2
2
b – 4ac = (10) – 4(3)(–12) a = 3, b = 10, and
To find accurate solutions, use the Quadratic c = –12
Formula. –b ± b – 4ac
2
t = = 244 Simplify.
2a
The discriminant is 244. Since
Thrown on Mars Answer the discriminant is positive, the
2
–10 ± 10 – 4(–1.85)(2) equation has two real roots.
=
2(–1.85)
–10 ± 114.8
= State the value of the discriminant for
–3.7 4x – 2x + 14 = 0. Then determine the number
2
t ≈ –0.19 or t ≈ 5.60 of real roots of the equation.

Thrown on Europa b – 4ac = (–2)² – 4(4)(14) a = 4, b = –2, and
2
c = 14
2
–10 ± 10 – 4(–0.65)(2)
=
2(–0.65) = –220 Simplify.
–10 ± 105.2
= The discriminant is –220. Since
–1.3
Answer the discriminant is negative, the
t ≈ –0.20 or t ≈ 15.6 equation has no real roots.

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