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Chapter 9: Problem 145

Determine the number of real solutions for each quadratic equation. (a) \(4 x^{2}-5 x+16=0\) (b) \(36 y^{2}+36 y+9=0\) (c) \(6 m^{2}+3 m-5=0\)

Short Answer

Expert verified
Equation (a) has no real solutions, equation (b) has one real solution, and equation (c) has two real solutions.

Step by step solution

01

- Understand the Discriminant

The number of real solutions for a quadratic equation of the form \(ax^2 + bx + c = 0\) is determined by its discriminant, which is given by \(D = b^2 - 4ac\). The discriminant tells us the nature of the roots: if \(D > 0\), there are two distinct real roots; if \(D = 0\), there is exactly one real root (a repeated root); if \(D < 0\), there are no real solutions (the roots are complex numbers).
02

- Apply the Discriminant to Equation (a)

For the quadratic equation \(4x^2 - 5x + 16 = 0\), identify \(a = 4\), \(b = -5\), and \(c = 16\). Plugging these values into the discriminant formula gives: \[D = (-5)^2 - 4(4)(16) = 25 - 256 = -231\]Since \(D < 0\), equation (a) has no real solutions.
03

- Apply the Discriminant to Equation (b)

For the quadratic equation \(36y^2 + 36y + 9 = 0\), identify \(a = 36\), \(b = 36\), and \(c = 9\). Plugging these values into the discriminant formula gives: \[D = 36^2 - 4(36)(9) = 1296 - 1296 = 0\]Since \(D = 0\), equation (b) has exactly one real solution.
04

- Apply the Discriminant to Equation (c)

For the quadratic equation \(6m^2 + 3m - 5 = 0\), identify \(a = 6\), \(b = 3\), and \(c = -5\). Plugging these values into the discriminant formula gives: \[D = 3^2 - 4(6)(-5) = 9 + 120 = 129\]Since \(D > 0\), equation (c) has two distinct real solutions.

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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 concept in understanding the nature of solutions for quadratic equations. Given a quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant (\( D \)) is calculated using the formula: \[ D = b^2 - 4ac \]. This simple calculation can provide crucial insight into the type and number of roots the equation has. The discriminant helps us determine if the quadratic equation has:
  • Two distinct real roots (\( D > 0 \))
  • One real root, also known as a repeated root (\( D = 0 \))
  • No real roots, implying the solutions are complex numbers ( \(D < 0 \)).
Understanding and calculating the discriminant is the first step in solving any quadratic equation.
Real Solutions
Real solutions of a quadratic equation are found when the discriminant (\( D \)) is greater than or equal to zero. Let's break down how the discriminant indicates the number of real solutions:
  • If \( D > 0 \), the quadratic equation has two distinct real solutions. This typically means the graph of the quadratic function will intersect the x-axis at two points.
  • If \( D = 0 \), the quadratic equation has exactly one real solution. In this case, the graph of the quadratic function touches the x-axis at just one point, referred to as a repeated root.
Finding real solutions, therefore, largely depends on calculating and interpreting the discriminant correctly.
Roots of Quadratic Equations
The roots of a quadratic equation are the values of the variable that satisfy the equation, where the quadratic expression equals zero. Roots can be found using the quadratic formula: \[x = \frac{-b \text{卤} \sqrt{D}}{2a}\]. Depending on the value of the discriminant (\( D \)), the formula provides different roots:
  • For \( D > 0 \), substituting the discriminant into the formula gives two distinct real roots.
  • For \( D = 0 \), the formula simplifies to yield one real root because the square root of zero is zero, eliminating the 卤 aspect.
  • For \( D < 0 \), the roots involve complex numbers because the square root of a negative number is imaginary.
Understanding how to find and interpret these roots is vital for solving quadratic equations thoroughly.
Nature of Roots
The nature of roots in a quadratic equation is closely tied to the discriminant (\( D \)). The nature, whether real or complex, distinct or repeated, is deduced by evaluating \( D \). Here's a quick recap:
  • When \( D > 0 \), the roots are real and distinct, meaning they are two different solutions.
  • When \( D = 0 \), the root is real and repeated, indicating the same solution occurs twice.
  • When \( D < 0 \), the roots are complex conjugates, featuring a pair of imaginary numbers expressed as \( a + bi \) and \( a - bi \).
Knowing the nature of roots is not just mathematical curiosity鈥� it's fundamental for graphing functions, solving equations, and understanding their behavior in various applications.

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Most popular questions from this chapter

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=x^{2}+4 x+3$$

Solve each inequality algebraically and write any solution in interval notation. $$-x^{2}+2 x-7 \geq 0$$

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Solve. Round answers to the nearest tenth. A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic function \(A(x)=x\left(50-\frac{x}{2}\right)\) gives the area, \(A\), of the dog run for the length, \(x,\) of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.

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