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Magazine Chapter 0: Problem 61
Solve using any method. $$t^{2}+5 t-6=0$$
Short Answer
Expert verified
The solutions are \( t = -6 \) and \( t = 1 \).
Step by step solution
01
Identify the Type of Equation
The given equation \( t^2 + 5t - 6 = 0 \) is a quadratic equation and is of the form \( at^2 + bt + c = 0 \) where \( a = 1 \), \( b = 5 \), and \( c = -6 \).
02
Check for Factoring
To solve the equation using factoring, look for two numbers that multiply to \( ac = -6 \) and add to \( b = 5 \). The numbers 6 and -1 satisfy these conditions because \( 6 \times (-1) = -6 \) and \( 6 + (-1) = 5 \).
03
Factor the Equation
Rewrite the middle term using the numbers found: \[t^2 + 6t - t - 6 = 0\]Group the terms:\[(t^2 + 6t) + (-t - 6) = 0\]Factor each group:\[t(t + 6) - 1(t + 6) = 0\]Factor out the common factor \((t + 6)\):\[(t + 6)(t - 1) = 0\]
04
Solve for the Variable
Set each factor equal to zero and solve:\[t + 6 = 0 \quad \Rightarrow \quad t = -6 \t - 1 = 0 \quad \Rightarrow \quad t = 1\]
05
Verify Solutions
Plug the solutions back into the original equation to verify they satisfy it.For \( t = -6 \):\[(-6)^2 + 5(-6) - 6 = 36 - 30 - 6 = 0\]For \( t = 1 \):\[1^2 + 5(1) - 6 = 1 + 5 - 6 = 0\]Both solutions are correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Method
The factoring method is one of the primary ways to solve quadratic equations. It involves expressing the quadratic equation in the form \( (x + p)(x + q) = 0 \). The idea is to find two binomials whose product gives back the original quadratic equation.To solve by factoring, follow these steps:
- Identify a, b, and c: First, confirm that the equation is quadratic by writing it in standard form \( ax^2 + bx + c = 0 \). Here, you'll identify the coefficients \( a \), \( b \), and \( c \).
- Find two numbers: Look for two numbers that multiply to \( ac \) (the product of \( a \) and \( c \)) and add up to \( b \).
- Rewrite the equation: Break the middle term into two terms using the numbers you found.
- Group and factor: Group the terms into two pairs and factor each pair separately. You should now have a common factor to factor out completely.
Quadratic Formula
Sometimes, the quadratic equation is not easily factored, or factoring might be challenging. In such cases, the quadratic formula serves as a reliable method to find the roots of any quadratic equation. This formula is given by:\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how you use it:
- Plug into the Formula: Substitute the values of \( a \), \( b \), and \( c \) from the equation \( ax^2 + bx + c = 0 \) into the formula.
- Calculate the Discriminant: The expression under the square root, \( b^2 - 4ac \), is called the discriminant. It tells you the nature of the roots.
- If it is positive, there are two real and distinct roots.
- If it is zero, there is one real root (also known as a repeated root).
- If it is negative, there are two complex roots.
Solving Quadratic Equations
Solving quadratic equations can be achieved through several methods, each with its own advantages depending on the context.
- The quadratic formula is reliable for any quadratic equation.
- Completing the square is useful for transforming equations into vertex form where the vertex or maximum/minimum point is needed.
- Graphing provides a visual interpretation but might not always offer precise solutions unless used with digital tools. Understanding each method allows flexibility in problem-solving and ensures that a suitable approach is taken based on the equation you need to solve.
- Factoring: As explained, this is effective for equations that are easily factored into binomials.
- Quadratic Formula: A universal tool for finding roots, especially when factoring is complex or impossible.
- Completing the Square: This method involves rewriting the equation into a perfect square trinomial, making it easy to solve by taking the square root of both sides.
- Graphing: By plotting the quadratic equation as a parabola, its roots, or x-intercepts, become visible as the points where the graph crosses the x-axis.
- The quadratic formula is reliable for any quadratic equation.
- Completing the square is useful for transforming equations into vertex form where the vertex or maximum/minimum point is needed.
- Graphing provides a visual interpretation but might not always offer precise solutions unless used with digital tools. Understanding each method allows flexibility in problem-solving and ensures that a suitable approach is taken based on the equation you need to solve.
