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Chapter 11: Problem 63

Solve by completing the square. Show your work. $$ t^{2}+8 t-3=0 $$

Short Answer

Expert verified
t = -4 + \sqrt{19} and t = -4 - \sqrt{19}.

Step by step solution

01

- Move Constant Term to the Other Side

Start by moving the constant term to the other side of the equation. This gives: \[ t^2 + 8t = 3 \]
02

- Complete the Square

To complete the square, add and subtract \((\frac{b}{2})^2\). Here, \(b = 8\), so \((\frac{8}{2})^2 = 16\). Add 16 to both sides of the equation: \[ t^2 + 8t + 16 = 3 + 16 \] This simplifies to: \[ t^2 + 8t + 16 = 19 \]
03

- Write the Left Side as a Binomial Square

Now, write the left side as a perfect square trinomial: \[ (t + 4)^2 = 19 \]
04

- Solve for t by Taking the Square Root

Take the square root of both sides: \[ t + 4 = \pm \sqrt{19} \]
05

- Solve for t

Isolate \(t\) by subtracting 4 from both sides: \[ t = -4 \pm \sqrt{19} \]Thus, the solutions are: \[ t = -4 + \sqrt{19} \] and \[ t = -4 - \sqrt{19} \]

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Key Concepts

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

quadratic equations
A quadratic equation is a type of polynomial equation of the form: \[ ax^2 + bx + c = 0 \] Here, 'a,' 'b,' and 'c' are coefficients, where 'a' is not equal to zero. Quadratic equations can appear often in real-life situations, such as projectile motion, area problems, and more.Quadratic equations are typically solved using one of three methods: factoring, using the quadratic formula, or completing the square.
Below, we will focus on completing the square to solve a specific quadratic equation.
perfect square trinomial
To solve a quadratic equation by completing the square, you must first form a perfect square trinomial on one side of the equation. A perfect square trinomial is an expression of the form:\[ (x + d)^2 \] Where 'd' is half the coefficient of the 'x' term.For example, consider the quadratic equation given as:\[ t^2 + 8t - 3 = 0 \] We want to isolate the terms involving 't' and complete the square.
Here鈥檚 how it is done:
  • Move the constant to the other side: \[ t^2 + 8t = 3 \]
  • Next, find the value to add and subtract to complete the square. Here, \[ b = 8 \], so \[ \left(\frac{8}{2}\right)^2 = 16 \]. Add 16 to both sides: \[ t^2 + 8t + 16 = 3 + 16 \].
  • Write the resulting expression as a binomial square: \[ (t + 4)^2 = 19 \]
This way, you have transformed the quadratic expression into a perfect square trinomial.
solving quadratic equations
Once you have your equation in the form of a perfect square trinomial, the next step is to solve for the variable.Continuing from the previous example: \[ (t + 4)^2 = 19 \] Take the square root of both sides to remove the square:\[ t + 4 = \pm \sqrt{19} \]This step simplifies the equation to two possible solutions:
  • \[ t + 4 = \sqrt{19} \]
  • \[ t + 4 = -\sqrt{19} \]
Finally, isolate 't' by subtracting 4 from each solution:
  • \[ t = -4 + \sqrt{19} \]
  • \[ t = -4 - \sqrt{19} \]
Thus, the solutions to the quadratic equation are\[-4 + \sqrt{19} \] and \[-4 - \sqrt{19} \].

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Most popular questions from this chapter

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (-4,-2) $$

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Solve. $$ x^{2}=100 $$
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