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Magazine Chapter 9: Problem 148
Determine the number of real solutions for each quadratic equation. (a) \(25 p^{2}+10 p+1=0\) (b) \(7 q^{2}-3 q-6=0\) (c) \(7 y^{2}+2 y+8=0\)
Short Answer
Expert verified
(a) 1 real solution. (b) 2 real solutions. (c) No real solutions.
Step by step solution
01
Identify the coefficients
For each quadratic equation, identify the coefficients a, b, and c in the standard form of a quadratic equation, which is given by: ax^2 + bx + c = 0. (a) For the equation 25p^2 + 10p + 1 = 0 the coefficients are: a = 25 b = 10 c = 1
02
Discriminant of the Quadratic Equation (a)
For equation (a), compute the discriminant 鈭� using the formula: 鈭� = b^2 - 4ac. 鈭� = 10^2 - 4(25)(1) 鈭� = 100 - 100 鈭� = 0.
03
Determine the Number of Real Solutions (a)
The discriminant 鈭� determines the number of real solutions: - If 鈭� > 0, there are 2 real solutions. - If 鈭� = 0, there is 1 real solution. - If 鈭� < 0, there are no real solutions. Since 鈭� = 0 for equation (a), there is 1 real solution.
04
Coefficient Identification for Equation (b)
For the quadratic equation 7q^2 - 3q - 6 = 0, the coefficients are: a = 7 b = -3 c = -6.
05
Discriminant of the Quadratic Equation (b)
For equation (b), compute the discriminant 鈭� using the formula: 鈭� = b^2 - 4ac. 鈭� = (-3)^2 - 4(7)(-6) 鈭� = 9 + 168 鈭� = 177.
06
Determine the Number of Real Solutions (b)
Since 鈭� = 177 for equation (b), 鈭� > 0 , so there are 2 real solutions.
07
Coefficient Identification for Equation (c)
For the quadratic equation 7y^2 + 2y + 8 = 0, the coefficients are: a = 7 b = 2 c = 8.
08
Discriminant of the Quadratic Equation (c)
For equation (c), compute the discriminant 鈭� using the formula: 鈭� = b^2 - 4ac. 鈭� = 2^2 - 4(7)(8) 鈭� = 4 - 224 鈭� = -220.
09
Determine the Number of Real Solutions (c)
Since 鈭� = -220 for equation (c), 鈭� < 0 , so there are no real solutions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
discriminant
In the context of quadratic equations, the discriminant is a value that helps determine the nature and number of solutions. The discriminant is calculated using the formula: \[ \text{鈭唥 = b^2 - 4ac \] where the values of \( a \), \( b \), and \( c \) are the coefficients from the standard form of a quadratic equation, \( ax^2 + bx + c = 0 \). Depending on the value of the discriminant, you can have:
- Two Real Solutions: If \( \text{鈭唥 > 0 \), the quadratic equation has two distinct real solutions.
- One Real Solution: If \( \text{鈭唥 = 0 \), there is exactly one real solution (it鈥檚 a repeated or double root).
- No Real Solutions: If \( \text{鈭唥 < 0 \), there are no real solutions, but rather two complex solutions.
coefficients
In any quadratic equation in the form \( ax^2 + bx + c = 0 \), the values \( a \), \( b \), and \( c \) are known as the coefficients. Identifying these coefficients is a critical first step when addressing quadratic equations.
- a (the leading coefficient): The coefficient of the term with \( x^2 \). This term establishes the equation's quadratic nature.
- b (the linear coefficient): The coefficient of the term with \( x \). This term gives rise to the linear part of the equation.
- c (the constant term): The standalone number without any variables attached to it. It shifts the graph up or down.
quadratic equation analysis
Analyzing quadratic equations involves a systematic process to determine the type and number of solutions. This process generally includes:
- Step 1: Identifying Coefficients: Recognize the values of \( a \), \( b \), and \( c \) from the equation.
- Step 2: Computing the Discriminant: Use the formula \( \text{鈭唥 = b^2 - 4ac \) to find the discriminant.
- Step 3: Interpreting the Discriminant: Assess the discriminant value to conclude the number and nature of the solutions.
- Example (a): The equation \( 25p^2 + 10p + 1 = 0 \) has coefficients \( a = 25 \), \( b = 10 \), and \( c = 1 \). The discriminant is \( \text{鈭唥 = 10^2 - 4(25)(1) = 0 \). Thus, it has 1 real solution.
- Example (b): The equation \( 7q^2 - 3q - 6 = 0 \) has coefficients \( a = 7 \), \( b = -3 \), and \( c = -6 \). The discriminant is \( \text{鈭唥 = (-3)^2 - 4(7)(-6) = 177 \). Thus, it has 2 real solutions.
- Example (c): The equation \( 7y^2 + 2y + 8 = 0 \) has coefficients \( a = 7 \), \( b = 2 \), and \( c = 8 \). The discriminant is \( \text{鈭唥 = 2^2 - 4(7)(8) = -220 \). Thus, it has no real solutions.
