Problem 42 - Contemporary Precalculus

Question

Use the quadratic formula to solve the equation. $$2 t^{2}+4 t+1=0$$

Step-by-Step Solution

Answer

Answer: The solutions for the given quadratic equation are \(t_1 = -1 + \frac{1}{2}\sqrt{2}\) and \(t_2 = -1 - \frac{1}{2}\sqrt{2}\).

Step 1: Identify the coefficients

In the given equation, \(2t^2 + 4t + 1 = 0\), identify the coefficients a, b, and c, where a is the coefficient of \(t^2\), b is the coefficient of t, and c is the constant term. In this case, we have a = 2, b = 4, and c = 1.

Step 2: Recall the quadratic formula

The quadratic formula is used to find the values of t for a quadratic equation in the form \(at^2 + bt + c = 0\). The quadratic formula is: $$t = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Step 3: Substitute the coefficients into the quadratic formula

Now substitute a = 2, b = 4, and c = 1 into the quadratic formula: $$t = \frac{-4\pm\sqrt{4^2-4(2)(1)}}{2(2)}$$

Step 4: Simplify the expression under the square root

Next, simplify the expression inside the square root: $$t = \frac{-4\pm\sqrt{16-8}}{4}$$

Step 5: Calculate the values of t

Finally, calculate the two possible values of t using the quadratic formula: $$t = \frac{-4\pm\sqrt{8}}{4}$$ Now, simplify the square root of 8, which can be written as 2 times the square root of 2: $$t_{1,2} = \frac{-4\pm 2\sqrt{2}}{4}$$ Divide both terms in the numerator by 4: $$t_1 = \frac{-4 + 2\sqrt{2}}{4} = -1 + \frac{1}{2}\sqrt{2}$$ $$t_2 = \frac{-4 - 2\sqrt{2}}{4} = -1 - \frac{1}{2}\sqrt{2}$$ So, the solutions are: $$t_1 = -1 + \frac{1}{2}\sqrt{2}$$ $$t_2 = -1 - \frac{1}{2}\sqrt{2}$$

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is fundamental in algebra. A quadratic equation can always be written in the form of \( ax^2 + bx + c = 0 \), where \( a \) is not zero. When facing an equation like this, the quadratic formula \( x = \frac{-b\pm\sqrt{b^2-4ac}}{2a} \) becomes a powerful tool to find the roots, which are the values of \( x \) that satisfy the equation. To apply the formula, first identify the coefficients - \( a \) for \( x^2 \) term, \( b \) for \( x \) term, and \( c \) for the constant term.

Once these are determined, substituting them into the formula accurately gives us the roots after a few simplification steps. It’s critical to maintain precision during simplification to ensure the correct roots are found, and making sure to perform operations step-wise can help prevent possible errors.

Quadratic Equation Coefficients

In a quadratic equation, the coefficients hold much significance. They are typically denoted as \( a \) for the \( x^2 \) term, \( b \) for the \( x \) term, and \( c \) as the constant. It’s worth noting that if \( a \) is zero, the equation is no longer quadratic. For our exercise, the equation \( 2t^2+4t+1=0 \) furnishes us with \( a=2 \) , \( b=4 \) , and \( c=1 \).

The accurate identification of these coefficients is necessary for solving the equation using the quadratic formula. They directly influence the discriminant \( b^2-4ac \) - which determines the nature of roots – and the ultimate solutions we derive through the formula.

Square Root Simplification

Simplifying the square root is an essential part of solving quadratic equations using the quadratic formula. After calculating the discriminant \( b^2-4ac \), it ends up under the square root. It’s where square root simplification comes into play. The goal is to express the square root of numbers as simply as possible, often by finding the largest square factor.

For instance, \( \sqrt{8} \) can be broken down into \( \sqrt{4 \times 2} \) which simplifies to \( 2\sqrt{2} \) because \( \sqrt{4} \) equals 2. In the context of our problem \( t = -4\pm\sqrt{8} \) simplifies to \( t = -4\pm 2\sqrt{2} \) after recognizing that 8 is 4 times 2 and \( \sqrt{4} \) is 2. This step is vital because it results in the most simplified form of the roots, therefore making them easier to comprehend and operate with in further mathematical procedures or real-world application.