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Chapter 1: Problem 68

Solve equation. \(2 x^{2}+x=6\)

Short Answer

Expert verified
The solutions to the equation are \(x = 1.5\) and \(x = -2\)

Step by step solution

01

Set the Equation to the Standard Form

The standard form of a quadratic equation is \(ax^{2} + bx + c = 0\), so first, subtract 6 from both sides to get \(2x^{2} + x - 6 = 0\)
02

Calculate the Discriminant

The discriminant, denoted as \(d\), can be found using \(b^{2} - 4ac\). After substituting \(a = 2\), \(b = 1\), and \(c = -6\) into the equation, the discriminant is \(1 - 4 * 2 * -6 = 49\)
03

Use the Quadratic Formula

The quadratic formula is \(x = (-b 卤 鈭歞) / (2a)\). Substituting into this equation gives \(x = (-1 卤 鈭�49) / (2 * 2)\). Therefore, the solutions to the equation are \(x = (-1 + 7) / 4 = 1.5\) and \(x = (-1 - 7) / 4 = -2\)

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Key Concepts

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

Discriminant
The discriminant is a critical component in the world of quadratic equations. It helps you determine the nature of the roots of the equation without actually solving it. The discriminant is denoted by the symbol \(d\) and can be calculated using the formula: \(b^2 - 4ac\). Here, \(a\), \(b\), and \(c\) are the coefficients from your quadratic equation in standard form.
For the equation \(2x^2 + x = 6\), once you've rewritten it in standard form as \(2x^2 + x - 6 = 0\), the coefficients are:
  • \(a = 2\)
  • \(b = 1\)
  • \(c = -6\)
Calculate the discriminant: - \(d = b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot (-6)\) - \(d = 1 + 48 = 49\) When the discriminant is positive, like 49 in this instance, you can expect two distinct real solutions.
Quadratic Formula
The quadratic formula is a powerful tool for finding the solutions to any quadratic equation. This formula is expressed as: \[x = \frac{-b \pm \sqrt{d}}{2a}\]where \(d\) is the discriminant. By using this formula, you can quickly find the roots of the quadratic equation once you know \(a\), \(b\), \(c\), and the discriminant.
For our example \(2x^2 + x - 6 = 0\), the quadratic formula becomes: - \(x = \frac{-1 \pm \sqrt{49}}{2 \cdot 2}\) This equation simplifies into two possible solutions for \(x\):
  • \(x = \frac{-1 + 7}{4} = 1.5\)
  • \(x = \frac{-1 - 7}{4} = -2\)
So, the solutions to this quadratic equation are \(x = 1.5\) and \(x = -2\).
Standard Form
The standard form of a quadratic equation is the expression \(ax^2 + bx + c = 0\). This form is crucial because it sets the foundation for applying both the quadratic formula and calculating the discriminant. Always aim to rearrange any quadratic equation into this format as your first step.
For the given exercise, you begin with \(2x^2 + x = 6\). By subtracting 6 from both sides, you convert it into the standard form: - \(2x^2 + x - 6 = 0\) This form clearly shows the coefficients \(a\), \(b\), and \(c\), making it easier to apply further algebraic techniques. The process of rearranging equations into the standard form will become second nature with practice.

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Most popular questions from this chapter

A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is \(x^{2}+y^{2}=25\) at the point \((3,-4)\)

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