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Magazine Chapter 11: Problem 19
Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ -2 t(t+2)=-3 $$
Short Answer
Expert verified
t = -1 + \frac{\sqrt{10}}{2} and t = -1 - \frac{\sqrt{10}}{2}
Step by step solution
01
Expand the Equation
First, expand the left-hand side by distributing \(t\): \(-2 t^2 - 4t = -3\).
02
Move All Terms to One Side
Add 3 to both sides to set the equation to zero: \(-2t^2 - 4t + 3 = 0\).
03
Identify Coefficients
Identify the coefficients \(a = -2, b = -4, c = 3\).
04
Apply the Quadratic Formula
The quadratic formula is given by \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the coefficients into the formula: \(t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-2)(3)}}{2(-2)} \).
05
Simplify the Equation
Simplify the math inside the sqrt and the fraction: \(t = \frac{4 \pm \sqrt{16 + 24}}{-4} \) which simplifies to \(t = \frac{4 \pm \sqrt{40}}{-4} \).
06
Simplify the Square Root
\(\sqrt{40} = 2\sqrt{10}\), so the equation becomes \(t = \frac{4 \pm 2\sqrt{10}}{-4} \).
07
Divide by the Denominator
Divide each term by -4: \(t = -1 \pm \frac{\sqrt{10}}{2} \).
08
State the Solution
The solutions are \(t = -1 + \frac{\sqrt{10}}{2}\) and \(t = -1 - \frac{\sqrt{10}}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
solving quadratic equations
Understanding how to solve quadratic equations is essential in algebra. A quadratic equation looks like this: \text{ax}^2 + \text{bx} + \text{c} = 0, where \text{a}, \text{b}, and \text{c} are coefficients. The goal is to find the values of the variable (like \text{t}) that make the equation true. To solve a quadratic equation, you have several methods:
- Factoring: This works when it's easy to find two numbers that multiply to give \text{ac} and add to give \text{b}.
- Completing the Square: This involves transforming the equation into a perfect square trinomial.
- Quadratic Formula: This formula can always be used, even when other methods are tricky. It's given by \text{\(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)}.
quadratic formula
The quadratic formula is a powerful tool to solve any quadratic equation. It stems from the general form of a quadratic equation. Here's a breakdown:
The quadratic formula: \text{\(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)}.
To use it, you need to:
For our problem, it looks like this: \(t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-2)(3)}}{2(-2)}\).
After substituting, simplify within the square root and in the fraction. This helps break down the equation step-by-step which is essential for clarity.
The quadratic formula: \text{\(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)}.
To use it, you need to:
- Identify the coefficients \text{a}, \text{b}, and \text{c}: In our example, \text{a = -2}, \text{b = -4}, and \text{c = 3}.
- Substitute these values into the quadratic formula:
For our problem, it looks like this: \(t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-2)(3)}}{2(-2)}\).
After substituting, simplify within the square root and in the fraction. This helps break down the equation step-by-step which is essential for clarity.
step-by-step algebra solutions
Breaking down the solution into clear steps is crucial for understanding. Here's a detailed walk-through:
Remember to check back and ensure each step is clear. This way, solving quadratic equations becomes much easier and systematic.
- Step 1: Expand the equation - Start by distributing variables. For \text{-2t(t + 2)}, you'd get \text{-2t^2 - 4t = -3}.
- Step 2: Move all terms to one side - To standardize, add 3 to both sides: \text{-2t^2 - 4t + 3 = 0}.
- Step 3: Identify coefficients - \text{a = -2}, \text{b = -4}, and \text{c = 3}.
- Step 4: Apply the quadratic formula - Substitute: \(t = \frac{4 \pm \sqrt{16 + 24}}{-4}\).
- Step 5: Simplify - Simplify under the square root to get \text{\(\sqrt{40}\)}, and further simplify: \(2\sqrt{10}\).
- Step 6: Divide by the denominator - Break down to: \(t = -1 \pm \frac{\sqrt{10}}{2}\).
- Step 7: State the solution - Finally, your solutions are: \(t = -1 + \frac{\sqrt{10}}{2}\) and \(t = -1 - \frac{\sqrt{10}}{2}\).
Remember to check back and ensure each step is clear. This way, solving quadratic equations becomes much easier and systematic.
