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Chapter 2: Problem 33

Solve the quadratic equation by completing the square. $$x^{2}+4 x=-3$$

Short Answer

Expert verified
The solutions to the equation are \(x = 0\) and \(x = -4\).

Step by step solution

01

Reorganize The Equation

To start, the equation should be arranged with x-terms on one side and the constant on the other. So, the equation becomes \(x^{2} + 4x + 3 = 0\)
02

Complete The Square

In order to complete the square, we need to add a value to both sides of the equation that makes the left side a perfect square trinomial. This value is \((b/2)^{2}\), here \(b\) is the coefficient of \(x\). In our case \(b = 4\), so the value we need to add is \((4/2)^{2} = 4\), which gives us a new equation: \(x^{2} + 4x + 4 = 0 + 4\) or simplified \(x^{2} + 4x + 4 = 4\)
03

Rewrite Left Side as a Square

The left side of the equation is now a perfect square trinomial. It can be written as \( (x + 2)^{2}\). Therefore, the equation now reads: \((x + 2)^{2} = 4\)
04

Solve for 'x'

Take the square root of both sides of the equation. This gives \(x + 2 = \pm 2\). This results in two solutions for 'x': \(x = 2 - 2 = 0\) and \(x = -2 - 2 = -4\).

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Key Concepts

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

Completing the Square
Completing the square is a method used to solve quadratic equations, making it a powerful tool in algebra.
This technique transforms a quadratic equation into a perfect square trinomial, which is an expression that can be easily factored as a square of a binomial.
Here's how it works:
  • First, ensure that the quadratic equation is in the standard form of \[ax^2 + bx + c = 0\].
  • Move any constant terms to the opposite side of the equation.
  • Find the term that will complete the square: Take half of the coefficient of \(x\), square it, and add this square to both sides of the equation.
  • This process turns the quadratic expression on one side of the equation into a perfect square trinomial.
By completing the square, you simplify the equation into a simple expression that reveals the roots directly, making it straightforward to solve.
Perfect Square Trinomial
A perfect square trinomial is a special kind of quadratic expression that can be expressed as the square of a binomial.
It takes the form of \((a+b)^2 = a^2 + 2ab + b^2\).
In the context of solving quadratic equations, once the left side of the equation is a perfect square trinomial, it becomes easier to work with.
Consider \(x^2 + 4x + 4\), which can be rewritten as \((x + 2)^2\).
This transformation outlines the structure and helps in simplifying the equation.
  • Identify the quadratic term, \(x^2\), the linear term, \(bx\), and use them to find the value \((b/2)^2\) to complete the square.
  • Rewriting the trinomial as a square of a binomial confirms that it is a perfect square trinomial.
Having the perfect square trinomial means you can directly equate it and solve for \(x\), which simplifies the process immensely.
Solving Equations
Solving quadratic equations involves finding the values of \(x\) that satisfy the equation, meaning the solutions or roots.
After completing the square and reaching a perfect square trinomial, solving becomes straightforward.
The following steps outline the process:
  • Rewrite the trinomial in the form of \((x + n)^2 = k\), where \(n\) and \(k\) are constants.
  • Take the square root of both sides, ensuring you consider both the positive and negative roots: \(x + n = \pm \sqrt{k}\).
  • Solve for \(x\) by isolating it on one side of the equation, which typically gives two solutions.
Using these steps guarantees that you find all possible solutions to the quadratic equation, demonstrating a clear path from completing the square to arriving at the final solutions for \(x\).

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Most popular questions from this chapter

Suppose the vertex of the parabola associated with a certain quadratic function is \((2,1),\) and another point on this parabola is (3,-1) (a) Find the equation of the axis of symmetry of the parabola. (b) Use symmetry to find a third point on the parabola. (c) Sketch the parabola.

The surface area of a sphere is given by \(A(r)=4 \pi r^{2},\) where \(r\) is in inches and \(A(r)\) is in square inches. The function \(C(x)=6.4516 x\) takes \(x\) square inches as input and outputs the equivalent result in square centimeters. Find \((C \circ A)(r)\) and explain what it represents.

The height of a ball after being dropped from a point 100 feet above the ground is given by \(h(t)=-16 t^{2}+100,\) where \(t\) is the time in seconds since the ball was dropped, and \(h(t)\) is in feet. (a) When will the ball be 60 feet above the ground? (b) When will the ball reach the ground? (c) For what values of \(t\) does this problem make sense (from a physical standpoint)?

In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Find an expression for \((f \circ f)(t)\), and give the domain of \(f \circ f\)

Is it true that \((f g)(x)\) is the same as \((f \circ g)(x)\) for any functions \(f\) and \(g ?\) Explain.
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