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Chapter 4: Problem 55

Solve. Find exact solutions. $$2 x^{2}+12=5 x$$

Short Answer

Expert verified
The exact solutions are \(x = \frac{5 + \sqrt{71}i}{4}\) and \(x = \frac{5 - \sqrt{71}i}{4}\).

Step by step solution

01

- Move All Terms to One Side

Subtract \(5x\) from both sides of the equation: \[2x^2 + 12 - 5x = 0\].
02

- Arrange in Standard Form

Rearrange the equation into the standard form of a quadratic equation: \[2x^2 - 5x + 12 = 0\].
03

- Apply the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For our equation, \(a = 2\), \(b = -5\), and \(c = 12\).
04

- Compute the Discriminant

Calculate the discriminant: \(b^2 - 4ac = (-5)^2 - 4(2)(12) = 25 - 96 = -71\).
05

- Determine the Nature of the Roots

Since the discriminant is negative (\(-71\)), the equation has no real solutions and only has complex solutions.
06

- Find the Complex Solutions

Using the quadratic formula and the discriminant: \(x = \frac{-(-5) \pm \sqrt{-71}}{2(2)}\), which simplifies to \(x = \frac{5 \pm \sqrt{71}i}{4}\).

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Key Concepts

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

solving quadratic equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They can often be solved using various methods like factoring, completing the square, or using the quadratic formula. In our exercise, we began by manipulating the equation into standard form by moving all terms to one side: \(2x^2 + 12 - 5x = 0\). This helps us identify the coefficients \(a\), \(b\), and \(c\).
Next, we rearranged it to \(2x^2 - 5x + 12 = 0\). This form makes it ready for applying the quadratic formula or other solving methods.
complex solutions in quadratic equations
When solving quadratic equations, the discriminant \(b^2 - 4ac\) helps in identifying the nature of the solutions. A negative discriminant, as we found \((-71)\) in our exercise, indicates that the quadratic equation does not have real solutions.
Instead, it has complex solutions. These solutions include the imaginary unit \(i\), which satisfies \(i^2 = -1\). So, for our equation, we derive the solutions as \(x = \frac{5 \pm \sqrt{71}i}{4}\). The \pm \ symbol indicates two solutions: one with addition and one with subtraction.
quadratic formula
The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is a powerful tool for solving quadratic equations. It provides the exact roots based on the coefficients from the standard form. For any quadratic \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are simply plugged into the formula.
In our equation \(2x^2 - 5x + 12 = 0\), we identified \(a = 2\), \(b = -5\), and \(c = 12\). Applying these to the quadratic formula gives us the roots in terms of real and imaginary parts when the discriminant is negative, as shown: \(x = \frac{5 \pm \sqrt{71}i}{4}\). This provides the complete set of solutions for the equation.

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Most popular questions from this chapter

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Solve. $$x^{5}+24 > 3 x^{3}+8 x^{2}$$
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