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Magazine Chapter 0: Problem 15
Solve the following equations using any method: \(4.9 t^{2}+2 t-20=0\)
Short Answer
Expert verified
The solutions are approximately \(t \approx 1.826\) and \(t \approx -2.245\).
Step by step solution
01
Identify the type of equation
The equation given is \(4.9t^2 + 2t - 20 = 0\). This is a quadratic equation because it is in the standard form \(ax^2 + bx + c = 0\), where \(a = 4.9\), \(b = 2\), and \(c = -20\).
02
Use the Quadratic Formula
To find the solutions of the quadratic equation, we can use the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values of \(a\), \(b\), and \(c\) from the equation into the formula.
03
Calculate the Discriminant
First, compute the discriminant \(b^2 - 4ac\): \[ b^2 - 4ac = (2)^2 - 4(4.9)(-20) \] Calculate each part step by step: - \(b^2 = 4\) - \(-4ac = -4 \times 4.9 \times -20 = 392\) Thus, the discriminant is \(4 + 392 = 396\).
04
Calculate the Roots using the Quadratic Formula
Now that we have the discriminant, substitute the values into the formula to find the roots: \[ t = \frac{-2 \pm \sqrt{396}}{9.8} \] - Calculate \(\sqrt{396} \approx 19.9\) - Find the two possible values for \(t\): 1. \(t_1 = \frac{-2 + 19.9}{9.8} \approx 1.826\) 2. \(t_2 = \frac{-2 - 19.9}{9.8} \approx -2.245\)
05
Present the Solutions
The solutions to the equation \(4.9t^2 + 2t - 20 = 0\) are approximately \(t \approx 1.826\) and \(t \approx -2.245\).
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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 used to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). Finding the roots of such equations can be tricky, especially when they don't factor easily or have complex solutions. The quadratic formula simplifies this process significantly:
- It provides a general solution for any quadratic equation.
- The formula is expressed as \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Discriminant
The discriminant is a key part of the quadratic formula and is given by the expression \(b^2 - 4ac\). It tells us vital information about the nature of the roots of the quadratic equation. Here's how you can interpret the discriminant:
- If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
- If \(b^2 - 4ac = 0\), there is exactly one real root (also known as a repeated or double root).
- If \(b^2 - 4ac < 0\), the roots are complex or imaginary, indicating no real roots.
Roots of Equation
The roots of a quadratic equation are the solutions that satisfy the equation \(ax^2 + bx + c = 0\). These roots can be found using the quadratic formula, as shown in our example.
- Once the discriminant is evaluated, substitute it under the square root in the quadratic formula.
- Solve for \(t\) by performing the operations indicated, which involves adding and subtracting the square root value from \(-b\), followed by dividing by \(2a\).
Polynomial Equation
A polynomial equation is an expression of the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0\), where the highest exponent \(n\) determines the degree of the polynomial. Quadratic equations are a special type of polynomial equation with a degree of 2, as exemplified by the given equation \(4.9t^2 + 2t - 20 = 0\).
- The highest power (\(t^2\)) tells us it is quadratic.
- The coefficients \(4.9\), \(2\), and \(-20\) provide the specific details necessary for solving the equation.
