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Chapter 9: Problem 90

Solve by completing the square. $$t^{2}-10 t-6=5$$

Short Answer

Expert verified
The solutions are t = 11 and t = -1.

Step by step solution

01

- Move the constant term to the other side

Start by moving the constant term (-6) to the right side of the equation to isolate the quadratic and linear terms. \[ t^2 - 10t - 6 = 5 \] Add 6 to both sides: \[ t^2 - 10t = 11 \]
02

- Identify and add the square completion term

Find the term that completes the square on the left side of the equation. Take half the coefficient of t, which is -10, and square it. Half of -10 is -5, and \[ (-5)^2 = 25 \] Add 25 to both sides of the equation: \[ t^2 - 10t + 25 = 11 + 25 \]\[ t^2 - 10t + 25 = 36 \]
03

- Write the left side as a square of a binomial

The left side of the equation can now be factored as a perfect square: \[ (t - 5)^2 = 36 \]
04

- Solve for t

Take the square root of both sides to solve for t: \[ \large t - 5 = \pm \sqrt{36} \]\[ t - 5 = \pm 6 \]Solve for t by isolating it: \[ t - 5 = 6 \]\[ t = 11 \]and \[ t - 5 = -6 \]\[ t = -1 \]

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Key Concepts

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

quadratic equations
Quadratic equations are polynomial equations of degree two. They typically have the form ax虏 + bx + c = 0, where a, b, and c are constants.

One of the key features of quadratic equations is they graph as parabolas, which can open either up or down depending on the sign of a.

When solving quadratic equations, various methods can be used:
  • Factoring
  • Using the quadratic formula
  • Completing the square
  • Graphing
One particularly accessible method is completing the square, which transforms the equation into a perfect square trinomial.

For instance, if we consider the equation t.
factoring
Factoring involves expressing a quadratic equation as a product of its binomial factors. This method is particularly useful when the quadratic can easily be decomposed.

For example, given the equation x虏 - 5x + 6 = 0, we can factor it into (x - 2)(x - 3) = 0. This allows us to solve for x by setting each binomial equal to zero.

However, not all quadratic equations can be factored easily. When direct factoring is difficult or impossible, other methods like completing the square or using the quadratic formula come in handy.
solving equations
Solving quadratic equations involves finding the values of the variable that satisfy the equation. Here's a breakdown of solving equations by completing the square:

    Identify the quadratic and linear terms.

    Move the constant term to the other side of the equation, like moving '-6' in t虏 - 10t - 6 = 5 to get t虏 - 10t = 11.

    Complete the square by adding the appropriate value: take half the coefficient of t, square it, and add to both sides. For coefficient '-10', half is '-5', and (-5)虏 = 25, giving t虏 - 10t + 25 = 36.

    Factor the left side as a binomial square, like (t - 5)虏 = 36to
.
    Solve by taking the square root of both sides, which leads to two possible solutions for t: t - 5 = 卤6leading to t = 11 and t = -1

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Most popular questions from this chapter

(a) solve graphically and (b) write the solution in interval notation. $$-x^{2}+2 x+24<0$$

Graph each function using a horizontal shift. $$f(x)=(x-2)^{2}$$

Graph each function using a vertical shift. $$h(x)=x^{2}-5$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=x^{2}-6 x+15$$

Graph each function using a vertical shift. $$f(x)=x^{2}-7$$
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