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Chapter 11: Problem 49

Solve each quadratic equation by completing the square. \(2 x^{2}+3 x-5=0\)

Short Answer

Expert verified
The solutions to the equation are \[x = -\frac{3}{4} 卤 \frac{\sqrt{59}}{4}\]

Step by step solution

01

Divide by the Coefficient of \(x^2\)

In the given equation, \(2 x^{2}+3 x-5=0\), 2 is the coefficient of \(x^2\). We need to make this coefficient equal to 1. So, divide the entire equation by 2 to get: \[x^{2}+\frac{3}{2}x-\frac{5}{2}=0\]
02

Rearrange to Leave a 'space'

Rearrange the equation to form a perfect square trinomial on the left side, leaving a 'space': \[x^{2}+\frac{3}{2}x= \frac{5}{2} \]
03

Complete the Square

To complete the square, you add the square of half the coefficient of x to both sides of the equation. Half of the coefficient of x in our equation is \(\frac{3}{4}\). Its square is \((\frac{3}{4})^{2}=\frac{9}{16}\). Add \(\frac{9}{16}\) to both sides: \[x^{2}+\frac{3}{2}x+(\frac{3}{4})^{2}= \frac{5}{2}+(\frac{3}{4})^{2}\] which simplifies to: \[(x + \frac{3}{4})^{2} = \frac{59}{16}\]
04

Find the Root

Taking square root on both sides yields: \[x + \frac{3}{4} = 卤 \frac{\sqrt{59}}{4} \] Solving for x gives the roots of the equation: \[x = -\frac{3}{4} 卤 \frac{\sqrt{59}}{4}\]

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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 polynomial equation of degree 2. Its general form is given by \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). Quadratic equations can appear in many real-world problems, such as projectile motion, area calculations, and optimization problems.
  • Standard Form: The equation should always be written in the form \(ax^2 + bx + c = 0\) before starting to solve it.
  • Unique Degree: The highest exponent is 2, which determines that this is indeed a quadratic equation.
  • Coefficient of \(x^2\): The term "quadratic" comes from the Latin word "quadrare", meaning square, referring to the \(x^2\) term.
When solving quadratic equations, various methods can be used, such as factoring, using the quadratic formula, or completing the square. Each method can provide the roots, or solutions, of the quadratic equation depending on the given situation. Ensuring the correct setup of the equation is a crucial first step.
Perfect Square Trinomial
To solve quadratic equations by completing the square, we often try to transform the equation into a perfect square trinomial. A perfect square trinomial is a special kind of quadratic expression that can be written in the form \((x + d)^2\) or \((x - d)^2\). Transforming a quadratic equation into this form makes it easier to solve by taking the square root.Here鈥檚 how you create a perfect square trinomial:
  • Identify the Middle Term: In the expression \(ax^2 + bx + c\), the middle term \(bx\) is key to creating a perfect square trinomial.
  • Divide and Square: Take half of the coefficient of \(x\), divide it by 2, and then square it.
  • Add and Balance: Add this squared value inside the equation to create a perfect square trinomial. Make sure to also adjust the right-hand side of the equation accordingly.
Perfect square trinomials simplify the equation to a format that is easy to solve with either positive or negative values, leading directly to finding the roots after completing these steps.
Roots of an Equation
The roots of an equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). For a quadratic equation, there can be two real roots, one real root, or two complex roots, depending on the discriminant \(b^2 - 4ac\).To find the roots using the "completing the square" method:
  • Transform to Perfect Square: After rearranging the equation into a perfect square trinomial, take the square root of both sides.
  • Solve for \(x\): Isolate \(x\) by solving \(x + d = 卤\sqrt{k}\) where \(k\) is the value obtained after taking the square root.
  • Two Solutions: You will arrive at two potential solutions due to the 卤 operator.
By following the process of completing the square, you can easily determine the roots of the equation, giving insights into where the graph of the quadratic equation intersects the x-axis, if and where it does.

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Most popular questions from this chapter

A piece of wire is 8 inches long. The wire is cut into two pieces and then each piece is bent into a square. Find the length of each piece if the sum of the areas of these squares is to be 2 square inches. (IMAGE CANNOT COPY)

Solve inequality using a graphing utility. \(\frac{x+2}{x-3} \leq 2\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I obtained \(-17\) for the discriminant, so there are two imaginary irrational solutions.

What is the discriminant and what information does it provide about a quadratic equation?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I complete the square for the binomial \(x^{2}+b x\) I obtain a different polynomial, but when I solve a quadratic equation by completing the square, I obtain an equation with the same solution set.
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