Chapter 10: Q 283. (page 1237)

In the following exercises, determine the number of solutions to each quadratic equation.
$(a) 9 x^{2} - 6 x + 1 = 0 (b) 3 y^{2} - 8 y + 1 = 0 (c) 7 m^{2} + 12 m + 4 = 0 (d) 5 n^{2} - n + 1 = 0$

Short Answer

Expert verified

Part (a) 1 solution

Part (b) 2 solutions

Part (c) 2 solutions

Part (d) No solution

Step by step solution

Part (a) Step 1. Given information.

The given equation is:

$9 x^{2} - 6 x + 1 = 0$

Part (a) Step 2. Determine the number of solutions.

$9 x^{2} - 6 x + 1 = 0$

The given equation is in standard form, identify a, b, and c.

$a = 9, b = - 6, c = 1$

Write in discriminant, and substitute the values ofa, b,andc:

localid="1653901256344" $b^{2} - 4 a c = {(- 6)}^{2} - 4 (9) (1) = 36 - 36 = 0$

As the discriminant is zero, there will be only one real solution for this equation.

Part (b) Step 1. Determine the number of solutions.

The given equation is:

$3 y^{2} - 8 y + 1 = 0$

The given equation is in standard form, identify a, b, and c.

$a = 3, b = - 8, c = 1$

Write in discriminant, and substitute the values ofa, b,andc:

$b^{2} - 4 a c = {(- 8)}^{2} - 4 (3) (1) = 64 - 12 = 52$

As the discriminant is positive, there will be two real solutions for this equation.

Part (c) Step 1. Determine the number of solutions.

The given equation is:

$7 m^{2} + 12 m + 4 = 0$

The given equation is in standard form, identify a, b, and c.

$a = 7, b = 12, c = 4$

Write in discriminant, and substitute the values ofa, b,andc:

localid="1657726795773" $b^{2} - 4 a c = 12^{2} - 4 (7) (4) = 144 - 112 = 32$

As the discriminant is positive, there will be two real solutions for this equation.

Part (d) Step 1. Determine the number of solutions.

The given equation is:

$5 n^{2} - n + 1 = 0$

The given equation is in standard form, identify a, b, and c.

$a = 5, b = - 1, c = 1$

Write in discriminant, and substitute the values ofa, b,andc:

$b^{2} - 4 a c = {(- 1)}^{2} - 4 (5) (1) = 1 - 20 = - 19$

As the discriminant is negative, there will be no real solution for this equation.

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In the following exercises, determine the number of solutions to each quadratic equation.
$(a) 9 x^{2} - 6 x + 1 = 0 (b) 3 y^{2} - 8 y + 1 = 0 (c) 7 m^{2} + 12 m + 4 = 0 (d) 5 n^{2} - n + 1 = 0$

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Determine the number of solutions.

Part (b) Step 1. Determine the number of solutions.

Part (c) Step 1. Determine the number of solutions.

Part (d) Step 1. Determine the number of solutions.

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In the following exercises, determine the number of solutions to each quadratic equation.a9x2−6x+1=0b3y2−8y+1=0c7m2+12m+4=0d5n2−n+1=0

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Determine the number of solutions.

Part (b) Step 1. Determine the number of solutions.

Part (c) Step 1. Determine the number of solutions.

Part (d) Step 1. Determine the number of solutions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Theoretical and Mathematical Physics

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Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

In the following exercises, determine the number of solutions to each quadratic equation.
$(a) 9 x^{2} - 6 x + 1 = 0 (b) 3 y^{2} - 8 y + 1 = 0 (c) 7 m^{2} + 12 m + 4 = 0 (d) 5 n^{2} - n + 1 = 0$