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Chapter 10: Problem 26

Solve the quadratic equation by completing the square. $$3 x^{2}-12 x=-7$$

Short Answer

Expert verified
The solutions for the equation are \(x = 2 卤 鈭�(\frac{5}{3})\)

Step by step solution

01

Rearrange the equation

Rearrange the quadratic equation to be in the standard form \(ax^{2} + bx + c = 0\). That gives us \(3 x^{2} - 12x + 7 = 0\)
02

Completing the square

Express the equation as a perfect square trinomial. Divide the equation by the coefficient of \(x^{2}\) which is 3, we get \(x^{2} - 4x + \frac{7}{3} = 0\). To complete the square, we add and subtract \((\frac{b}{2})^2\) inside the equation, and get \((x - 2)^{2} - 4 + \frac{7}{3} = 0\)
03

Solve for x

Now use the square root properties \( (x - a)^{2} = b => x = a 卤 鈭歜 \) to solve for x. So the solutions will be \(x = 2 卤 鈭�(\frac{5}{3})\)

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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, particularly when they are not easily factorable. It involves transforming a quadratic expression into a perfect square trinomial. This allows us to simplify solving the equation.

Here's how completing the square works in a simple sense:
  • First, ensure the quadratic equation is in standard form: \(ax^2 + bx + c = 0\).
  • Divide every term by \(a\) if \(a\) is not 1, to simplify the equation.
  • Take half the coefficient of the \(x\)-term, square it, and add this square inside the equation to form a perfect square trinomial.
This method is based on the identity \((x + d)^2 = x^2 + 2dx + d^2\). Completing the square offers a clear path to move forward with solving the equation.
Perfect Square Trinomial
A perfect square trinomial is an expression that can be written as the square of a binomial. When dealing with quadratic equations, converting them into perfect square trinomials simplifies the process of finding the roots.

To create a perfect square trinomial:
  • Identify the \(b\) term in \(x^2 + bx + c = 0\).
  • Take half of \(b\), square it, and add it to both sides of the equation.
  • This transforms the expression into \((x + p)^2\) or \((x - p)^2\).
For example, given \(x^2 - 4x\),
  • take half of \(-4\), which is \(-2\),
  • square it to get \(4\),
  • and add and subtract \(4\) to achieve \((x - 2)^2\).
Recognizing these patterns is crucial for simplifying quadratic equations effectively.
Solving Quadratic Equations
Solving quadratic equations can be straightforward, especially once the equation is expressed in the form of a perfect square. There are generally three techniques for solving these equations: factoring, using the quadratic formula, or completing the square.

When a quadratic is in the form \((x + p)^2 = q\):
  • You can quickly take the square root of both sides.
  • This leads to two possibilities: \(x + p = \sqrt{q}\) or \(x + p = -\sqrt{q}\).
When solving \(3x^2 - 12x = -7\) by completing the square, for instance, you adjust the equation to create compact expressions that can readily be solved using these properties. This method of simplifying is powerful, offering a dependable strategy for resolving quadratic problems.
Square Root Properties
Utilizing square root properties effectively can greatly assist in solving equations involving squares. Once the equation is expressed as a perfect square, you can employ these properties to isolate and solve for \(x\).

The key properties to remember are:
  • For any real number \(n\), \(\sqrt{n^2} = |n|\).
  • If \((x - a)^2 = b\), then \(x - a\) can equal both \(\sqrt{b}\) and \(-\sqrt{b}\).
After using the completing the square method resulting in an expression like \((x - 2)^2 = \frac{5}{3}\), apply these properties to determine \(x\):
  • Adding and subtracting the square root of the constant yields the two potential solutions for \(x\).
  • This dual solution is due to the symmetrical nature of squares.
Remember, mastering square root properties is essential for accurately solving quadratic equations when they come to this stage.

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Most popular questions from this chapter

Perform the operation(s). $$(a+b i)-(a-b i)$$

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Vocabulary Explain the meanings of the terms domain and range in the context of a function.
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