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Chapter 9: Problem 131

Solve by using the Quadratic Formula. $$8 x^{2}-6 x+2=0$$

Short Answer

Expert verified
\(x = \frac{3 \, \pm \, \sqrt{7}i}{8}\)

Step by step solution

01

Identify the Coefficients

First, identify the coefficients from the quadratic equation in standard form: \(ax^2 + bx + c = 0\). For the equation \(8x^2 - 6x + 2 = 0\), the coefficients are: \(a = 8\), \(b = -6\), \(c = 2\).
02

Quadratic Formula

The quadratic formula is \(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\). We will use this formula to find the values of \(x\).
03

Calculate the Discriminant

Calculate the discriminant \(\Delta\) using the formula \(\Delta = b^2 - 4ac\). Here, it is \((-6)^2 - 4 \cdot 8 \cdot 2\): \(36 - 64 = -28\).
04

Plug into the Quadratic Formula

Substitute \(a\), \(b\), and \(\Delta\) into the quadratic formula: \(x = \frac{-(-6) \, \pm \, \sqrt{-28}}{2 \cdot 8}\). This simplifies to \(x = \frac{6 \, \pm \, \sqrt{-28}}{16}\).
05

Simplify the Expression

Since \(\sqrt{-28} = \sqrt{28}i\), the expression becomes \(x = \frac{6 \, \pm \, 2\sqrt{7}i}{16}\), which simplifies further to \(x = \frac{3 \, \pm \, \sqrt{7}i}{8}\).

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Key Concepts

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

coefficients
In a quadratic equation, coefficients are the numerical values that multiply the variables. For example, in the quadratic equation \(8x^2 - 6x + 2 = 0\), the coefficients are essential to solving the equation using the Quadratic Formula.
Let's break it down:
  • \(a\): The coefficient of \(x^2\), which is 8.
  • \(b\): The coefficient of \(x\), which is -6.
  • \(c\): The constant term, which is 2.
Identifying these coefficients correctly is the first step in applying the Quadratic Formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Without accurate coefficients, you can't solve the equation correctly.
discriminant
The discriminant helps determine the nature of the roots of a quadratic equation. It's the part of the Quadratic Formula under the square root: \(\Delta = b^2 - 4ac\).
For the equation \(8x^2 - 6x + 2 = 0\):
  • First, calculate \(b^2\): \((-6)^2 = 36\).
  • Next, calculate \(4ac\): \(4 \cdot 8 \cdot 2 = 64\).
  • Finally, find the discriminant by subtracting \(4ac\) from \(b^2\): \(36 - 64 = -28\).
If \(\Delta\) is negative, like in this case, the quadratic equation has complex roots, meaning they include imaginary numbers.
complex numbers
Complex numbers come into play when dealing with a negative discriminant. Complex numbers involve an imaginary unit represented as \(i\), where \(i\) is defined as \(\sqrt{-1}\). This means if you have \(\sqrt{-28}\), it's the same as \(\sqrt{28}i\).
In our example, the equation simplifies to:
  • First, acknowledge \(\sqrt{-28}\) as \(\sqrt{28}i\).
  • Rewrite the quadratic formula with this substitution: \(x = \frac{6 \pm \sqrt{28}i}{16}\).
  • Simplify further by reducing to lowest terms: \(x = \frac{3 \pm \sqrt{7}i}{8}\).
This provides you with the complex roots of the given quadratic equation.
standard form
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). It's a way to ensure all terms are on one side of the equation, set equal to zero.
For example, to solve \(8x^2 - 6x + 2 = 0\) using the Quadratic Formula, make sure it's in standard form where:
  • \(a = 8\)
  • \(b = -6\)
  • \(c = 2\)
The standard form makes it straightforward to identify the coefficients \(a\), \(b\), and \(c\) correctly, which are essential for solving the equation using the Quadratic Formula. Always remember to rearrange any given quadratic equation to its standard form before proceeding with other steps.

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Most popular questions from this chapter

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=-x^{2}+2 x-7$$

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