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Magazine Chapter 1: Problem 4
Solve the following quadratic equations using the quadratic formula: (a) \(x^{2}+x-1=0\) (b) \(t^{2}-3 t-2=0\)(c) \(h^{2}+5 h+1=0\) (d) \(0.5 x^{2}+3 x-2=0\) (e) \(2 k^{2}-k-3=0\) (f) \(-y^{2}+3 y+1=0\) (g) \(3 r^{2}=7 r+2\) (h) \(x^{2}-70=0\) (i) \(4 s^{2}-2=s\) (j) \(2 t^{2}+5 t+2=0\) (k) \(3 x^{2}=50\)
Short Answer
Expert verified
Use the quadratic formula: solutions are given by \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), resulting in unique solutions for each equation in the forms described in the steps.
Step by step solution
01
Understand the Quadratic Formula
The quadratic formula is used to find the solutions of a quadratic equation in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula will be applied to each of the given equations.
02
Equation (a): Identify Parameters
For the equation \( x^2 + x - 1 = 0 \), identify the parameters: \( a = 1 \), \( b = 1 \), \( c = -1 \). These parameters will be used in the quadratic formula.
03
Equation (a): Calculate Discriminant
Calculate the discriminant for the equation: \( b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-1) = 1 + 4 = 5 \).
04
Equation (a): Apply the Quadratic Formula
Substitute \( a \), \( b \), \( c \), and the discriminant into the quadratic formula: \( x = \frac{-1 \pm \sqrt{5}}{2} \).
05
Equation (b): Identify Parameters
For the equation \( t^2 - 3t - 2 = 0 \), identify \( a = 1 \), \( b = -3 \), \( c = -2 \).
06
Equation (b): Calculate Discriminant
Calculate the discriminant: \( b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot (-2) = 9 + 8 = 17 \).
07
Equation (b): Apply the Quadratic Formula
Calculate \( t \) using the formula: \( t = \frac{3 \pm \sqrt{17}}{2} \).
08
Equation (c): Identify Parameters
For \( h^2 + 5h + 1 = 0 \), identify \( a = 1 \), \( b = 5 \), \( c = 1 \).
09
Equation (c): Calculate Discriminant
Calculate the discriminant: \( b^2 - 4ac = 5^2 - 4 \cdot 1 \cdot 1 = 25 - 4 = 21 \).
10
Equation (c): Apply the Quadratic Formula
Find \( h \) using the formula: \( h = \frac{-5 \pm \sqrt{21}}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool in algebra for solving quadratic equations. A quadratic equation is any equation that can be rearranged in the form: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero.
The quadratic formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula allows you to find the values of \( x \) that satisfy the equation. These values are called the roots of the quadratic equation.
To use the quadratic formula, follow these easy steps:
The quadratic formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula allows you to find the values of \( x \) that satisfy the equation. These values are called the roots of the quadratic equation.
To use the quadratic formula, follow these easy steps:
- Identify the values of \( a \), \( b \), and \( c \) in the equation.
- Substitute these values into the quadratic formula.
- Compute the expression under the square root, known as the discriminant.
- Calculate the two possible values for \( x \) by using both the plus (\( + \)) and minus (\( - \)) signs before the square root.
Discriminant
The discriminant is a crucial part of the quadratic formula, helping us determine the nature and number of roots of a quadratic equation. The discriminant is given by the expression: \[ b^2 - 4ac \].
Here is how to understand the discriminant:
This insight is particularly useful in real-world applications and higher-level math problem solving.
Here is how to understand the discriminant:
- If the discriminant is positive, the quadratic equation has two distinct real roots.
- If the discriminant is zero, the quadratic equation has exactly one real root, known as a repeated root.
- If the discriminant is negative, the quadratic equation has no real roots, but two complex roots.
This insight is particularly useful in real-world applications and higher-level math problem solving.
Math Problem Solving
Solving math problems, especially quadratic equations, can seem complex, but breaking them down into manageable steps simplifies the process. Here's a general guide to math problem solving for quadratic equations:
- **Understand the problem**: Clearly identify what the problem is asking for. In this case, finding the roots of the quadratic equations.
- **Translate the problem**: Write the problem in a familiar mathematical form, such as \( ax^2 + bx + c = 0 \).
- **Identify the process**: Since you are dealing with quadratics, plan to use the quadratic formula.
- **Execute the process step-by-step**: Substituting the values into the formula, calculating the discriminant, and solving for the roots.
- **Check your solution**: Verify if your solutions satisfy the original equation. This step ensures accuracy and understanding.
Polynomials
Polynomials are algebraic expressions that consist of variables and coefficients. A quadratic equation is a specific type of polynomial, with the highest exponent of the variable being 2. Understanding polynomials is essential in mastering quadratic equations.
Here are some basic elements of polynomials:
Here are some basic elements of polynomials:
- The **degree** of a polynomial is the highest power of the variable. In quadratics, this degree is always 2.
- **Terms** are the parts of the expression separated by plus or minus signs. For instance, in \( x^2 + x - 1 \), there are three terms.
- **Coefficients** are the numbers in front of the variables. In \( ax^2 + bx + c \), \( a \), \( b \), and \( c \) are the coefficients.
