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Chapter 10: Problem 134

In the following exercises, determine the number of solutions to each quadratic equation. a. \(25 p^{2}+10 p+1=0\) b.\(7 q^{2}-3 q-6=0\) c.\(7 y^{2}+2 y+8=0\) d.\(25 z^{2}-60 z+36=0\)

Short Answer

Expert verified
a) 1 solutionb) 2 solutionsc) 0 solutionsd) 1 solution

Step by step solution

01

- Identify the Quadratic Equation

Recognize the quadratic equation in standard form: a) 25 p^{2}+10 p+1=0 b) 7 q^{2}-3 q-6=0 c) 7 y^{2}+2 y+8=0 d) 25 z^{2}-60 z+36=0
02

- Calculate the Discriminant

Use the discriminant formula \(\b = b^2 - 4ac\) where a, b, and c are coefficients from the quadratic equations. Calculate it for each equation: a) Discriminant for 25 p^{2}+10 p+1=0: \(b = 10\) \(a = 25\) \(c = 1\) \(\b = 10^2 - 4(25)(1) = 100 - 100 = 0\) b) Discriminant for 7 q^{2}-3 q-6=0: \(b = -3\) \(a = 7 \) \(c = -6\) \(\b = (-3)^2 - 4(7)(-6) = 9 + 168 = 177\) c) Discriminant for 7 y^{2}+2 y+8=0: \(b = 2 \) \(a = 7 \) \(c = 8\) \(\b = 2^2 - 4(7)(8) = 4 - 224 = -220\) d) Discriminant for 25 z^{2}-60 z+36=0: \(b = -60\) \(a = 25 \) \(c = 36\) \(\b = (-60)^2 - 4(25)(36) = 3600 - 3600 = 0\)
03

- Determine the Number of Solutions

Interpret the discriminant results: - If \(\b > 0\), the equation has two real solutions. - If \(\b = 0\), the equation has one real solution. - If \(\b < 0\), the equation has no real solutions. Applying the interpretation: a) The discriminant is 0, so equation (a) has one real solution. b) The discriminant is 177, so equation (b) has two real solutions. c) The discriminant is -220, so equation (c) has no real solution. d) The discriminant is 0, so equation (d) has one real solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discriminant
The discriminant is a key element in understanding quadratic equations. It helps us determine the number and type of solutions. The discriminant is found using the formula: \(\Delta = b^2 - 4ac \), where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation in its standard form. To remember:
  • Identify the coefficients: \(a\), \(b\), and \(c\).
  • Plug them into the discriminant formula: \(\Delta = b^2 - 4ac \).
  • Calculate the value.
The value of the discriminant tells us whether the quadratic equation has one, two, or no real solutions.
Number of Solutions
In a quadratic equation, the discriminant helps us determine the number of solutions. Here's what it tells us:
If \(\Delta > 0\), there are two distinct real solutions.
If \(\Delta = 0\), there is exactly one real solution.
If \(\Delta < 0\), there are no real solutions.
To understand this better with examples:
  • For \(25 p^{2}+10 p+1=0\), \(\Delta = 0\), so it has one real solution.
  • For \(7 q^{2}-3 q-6=0\), \(\Delta = 177\), so it has two real solutions.
  • For \(7 y^{2}+2 y+8=0\), \(\Delta = -220\), so it has no real solutions.
  • For \(25 z^{2}-60 z+36=0\), \(\Delta = 0\), so it has one real solution.
By interpreting the discriminant, we can predict and understand the roots of the quadratic equation without actually solving it.
Standard Form
Understanding the standard form of a quadratic equation is essential. This form is \(ax^2 + bx + c = 0 \).
  • \(a\): coefficients of \(x^2\)
  • \(b\): coefficients of \(x\)
  • \(c\): constant term
Recognizing a quadratic equation in this format helps you to quickly identify the coefficients. For example:
  • For \(25p^{2}+10p+1=0\), \(a=25\), \(b=10\), \(c=1\).
  • For \(7q^{2}-3q-6=0\), \(a=7\), \(b=-3\), \(c=-6\).
  • For \(7y^{2}+2y+8=0\), \(a=7\), \(b=2\), \(c=8\).
  • For \(25z^{2}-60z+36=0\), \(a=25\), \(b=-60\), \(c=36\).
Identifying these coefficients is your first step to using the discriminant effectively.

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Most popular questions from this chapter

In the following exercises, solve. Round answers to the nearest tenth. A stone is thrown vertically upward from a platform that is 20 feet high at a rate of \(160 \mathrm{ft} / \mathrm{sec}\). Use the quadratic equation \(h=-16 t^{2}+160 t+20\) to find how long it will take the stone to reach its maximum height, and then find the maximum height.

In the following exercises, find (a) the axis of symmetry and (b) the vertex. $$ y=x^{2}+8 x-1 $$

In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. $$ y=3 x^{2}-6 x-1 $$

The length of a rectangular driveway is five feet more than three times the width. The area is 350 square feet. Find the length and width of the driveway.

The product of two consecutive odd numbers is \(483 .\) Find the numbers.
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