Problem 134 - Elementary Algebra

Question

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.\(25 z^{2}-60 z+36=0\)

Step-by-Step Solution

Answer

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

Step 1 - 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

Step 2 - 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: \(b = -60\) \(a = 25 \) \(c = 36\) \(\b = (-60)^2 - 4(25)(36) = 3600 - 3600 = 0\)

Step 3 - 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.

Key Concepts

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.