Problem 70
Question
Solve using the quadratic formula. $$ -8 x_{2}+20 x-13=0 $$
Step-by-Step Solution
Verified Answer
The roots are \(x = \frac{5}{4} - \frac{i}{4}\) and \(x = \frac{5}{4} + \frac{i}{4}\).
1Step 1: Identify the Coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). In the given equation \(-8x^2 + 20x - 13 = 0\), identify \(a = -8\), \(b = 20\), and \(c = -13\).
2Step 2: Write the 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 \(x\).
3Step 3: Calculate the Discriminant
Calculate the discriminant using the formula \( b^2 - 4ac \). Substitute \(a = -8\), \(b = 20\), and \(c = -13\):\[ 20^2 - 4(-8)(-13) = 400 - 416 = -16 \].
4Step 4: Analyze the Discriminant
The discriminant is \(-16\), which is less than zero. This means that the quadratic equation has no real roots, but two complex roots.
5Step 5: Use the Quadratic Formula for Complex Roots
Since the discriminant is negative, compute the square root of \(-16\) as \(4i\), where \(i\) is the imaginary unit. Substitute into the quadratic formula:\[ x = \frac{-20 \pm 4i}{-16} \].
6Step 6: Simplify the Expression
Simplify both parts of the expression:\[ x = \frac{-20 + 4i}{-16} = \frac{5}{4} - \frac{i}{4} \] and \[ x = \frac{-20 - 4i}{-16} = \frac{5}{4} + \frac{i}{4} \]. Thus, the roots are \( x = \frac{5}{4} - \frac{i}{4} \) and \( x = \frac{5}{4} + \frac{i}{4} \).
Key Concepts
Quadratic EquationComplex RootsDiscriminantImaginary Unit
Quadratic Equation
Quadratic equations are fundamental in algebra and often appear in the form of \(ax^2 + bx + c = 0\). These equations involve a degree of two, indicated by the square term, and can be solved using various methods like factoring, completing the square, or the quadratic formula.
In our example given by the equation \(-8x^2 + 20x - 13 = 0\), we identify the coefficients
In our example given by the equation \(-8x^2 + 20x - 13 = 0\), we identify the coefficients
- \(a = -8\)
- \(b = 20\)
- \(c = -13\)
Complex Roots
When solving a quadratic equation with a negative discriminant, the roots of the equation are known as complex roots. They incorporate the imaginary unit, denoted by \(i\), which represents the square root of \(-1\).
In the case of our quadratic equation \(-8x^2 + 20x - 13 = 0\), the discriminant is computed as \(-16\), indicating complex solutions. The roots in this scenario are \( \frac{5}{4} + \frac{i}{4} \) and \( \frac{5}{4} - \frac{i}{4} \), forming a pair of conjugates. Understanding complex roots is essential in mathematics because they expand the possibility of solutions beyond the real number line.
In the case of our quadratic equation \(-8x^2 + 20x - 13 = 0\), the discriminant is computed as \(-16\), indicating complex solutions. The roots in this scenario are \( \frac{5}{4} + \frac{i}{4} \) and \( \frac{5}{4} - \frac{i}{4} \), forming a pair of conjugates. Understanding complex roots is essential in mathematics because they expand the possibility of solutions beyond the real number line.
Discriminant
The discriminant is a key part of the quadratic formula, represented as \(b^2 - 4ac\). It helps determine the nature of the roots of a quadratic equation quickly.
For the equation \(-8x^2 + 20x - 13 = 0\), using
For the equation \(-8x^2 + 20x - 13 = 0\), using
- \(a = -8\)
- \(b = 20\)
- \(c = -13\)
Imaginary Unit
The imaginary unit, represented by \(i\), is a mathematical concept used to extend our number system into complex numbers. It is defined by the property that \(i^2 = -1\). This is a fundamental concept when dealing with quadratic equations that have complex roots.
In our example, after computing the discriminant as negative, we illustrated that the square root of \(-16\) is actually \(4i\). Consequently, \(i\) helps us express these roots. Complex numbers formed using the imaginary unit have the form \(a + bi\), where
In our example, after computing the discriminant as negative, we illustrated that the square root of \(-16\) is actually \(4i\). Consequently, \(i\) helps us express these roots. Complex numbers formed using the imaginary unit have the form \(a + bi\), where
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