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Chapter 11: Problem 43

Solve the equation and round off your answers to the nearest hundredth. $$t^{2}-7 t=10$$

Short Answer

Expert verified
The solutions are approximately 8.22 and -1.22.

Step by step solution

01

Move all terms to one side

Rewrite the equation by moving all terms to one side so it equals zero: \[ t^2 - 7t - 10 = 0 \]
02

Identify coefficients for the quadratic formula

In the quadratic equation \( t^2 - 7t - 10 = 0 \), identify the coefficients: \[ a = 1, \, b = -7, \, c = -10 \]
03

Apply the quadratic formula

Use the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the coefficients: \[ t = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-10)}}{2(1)} \]
04

Simplify under the square root

Calculate the value under the square root: \[ t = \frac{7 \pm \sqrt{49 + 40}}{2} \] Simplify further: \[ t = \frac{7 \pm \sqrt{89}}{2} \]
05

Calculate the two solutions

Find the two solutions by solving for both the plus and minus options: \[ t_1 = \frac{7 + \sqrt{89}}{2} \] \[ t_1 \approx \frac{7 + 9.43}{2} \approx 8.215 \approx 8.22 \] \[ t_2 = \frac{7 - \sqrt{89}}{2} \] \[ t_2 \approx \frac{7 - 9.43}{2} \approx -1.215 \approx -1.22 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic formula
To solve quadratic equations, the quadratic formula is indispensable. This powerful tool is expressed as: ewlineewlineo ewlineewline ewline \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]ewline ewline Here, \( t \) represents the variable we are solving for. ewline ewline The coefficients a, b, and c come from the standard form of a quadratic equation: ewline \[ at^{2} + bt + c = 0 \]ewline
rounding numbers
Rounding numbers is essential for simplification and relatability in real-world contexts. In our specific example, we are tasked with rounding to the nearest hundredth. Here's how:
ewline When you get a solution like 8.215, you observe the thousandth place, which is 5 in this case. Since it is 5 or more, you round the hundredth place up by 1, making it 8.22.
ewline For the second solution, approximately -1.215, the same principle applies. We observe the thousandth place, which is again 5, so we round the hundredth place from 1 to 2, making it -1.22.
ewline By rounding to the nearest hundredth, you make the values more manageable without losing significant precision.
identifying coefficients
Before you can use the quadratic formula, you need to identify the coefficients in your quadratic equation. The standard form for a quadratic equation is: ewline \[ at^{2} + bt + c = 0 \]ewline ewline This equation shows us where to find our coefficients. In our example, the equation is: ewline \[ t^2 - 7t - 10 = 0 \]ewline ewline From this: ewlineewline
  • \(a \ = 1 \ \)
  • \(b \ = -7 \ \)
  • \(c \ = -10 \ \)
ewline
By identifying these coefficients correctly, we can insert them into the quadratic formula and proceed with solving the equation.

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Most popular questions from this chapter

Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=x^{2}+3 x+2$$

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