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Magazine Chapter 6: Problem 98
Solve each equation. $$ -8 x^{2}+3-10 x=0 $$
Short Answer
Expert verified
The solutions are \(x = -1.5\) and \(x = \frac{1}{4}\).
Step by step solution
01
Identify the type of equation
The equation \(-8x^2 + 3 - 10x = 0\) is a quadratic equation. It is in the form \(ax^2 + bx + c = 0\), where \(a = -8\), \(b = -10\), and \(c = 3\).
02
Rearrange the equation
Rearrange the terms into standard quadratic form, \(-8x^2 - 10x + 3 = 0\). This helps in identifying the coefficients clearly.
03
Calculate the discriminant
Use the formula for the discriminant \(D = b^2 - 4ac\). Substitute the values: \(D = (-10)^2 - 4(-8)(3) = 100 + 96 = 196\).
04
Determine the nature of the roots
Since the discriminant \(D = 196\) is positive, the quadratic equation has two distinct real roots.
05
Apply the quadratic formula
The quadratic formula is given by \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute the known values: \(-b = 10\), \( \sqrt{196} = 14\), and \(2a = -16\). This gives the equation \(x = \frac{10 \pm 14}{-16}\).
06
Solve for the roots
Calculate the solutions: 1. \(x_1 = \frac{10 + 14}{-16} = \frac{24}{-16} = -1.5\).2. \(x_2 = \frac{10 - 14}{-16} = \frac{-4}{-16} = \frac{1}{4}\).
07
Verify the solutions
Substitute \(x_1 = -1.5\) and \(x_2 = \frac{1}{4}\) back into the original equation to ensure they satisfy \(-8x^2 + 3 - 10x = 0\). When checked, both values satisfy the equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Quadratic Formula
The quadratic formula is a powerful tool used to solve any quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\). This formula is especially useful when factoring the equation is difficult or impossible. The quadratic formula is expressed as follows:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's how it works:
- The coefficients \(a\), \(b\), and \(c\) represent the respective numbers in front of \(x^2\), \(x\), and the constant term in the equation.
- Once \(a\), \(b\), and \(c\) are identified, they are simply plugged into the formula.
- This formula includes the symbol \(\pm\), which indicates two possible solutions: one using \(+\) and the other using \(-\).
Decoding the Discriminant
The discriminant is a specific part of the quadratic formula that provides information about the nature of the roots of a quadratic equation. It is found under the square root symbol in the quadratic formula, represented by \(b^2 - 4ac\). Here鈥檚 how you can interpret the discriminant:
- If the discriminant \(D\) is greater than zero, the quadratic equation has two distinct real roots.
- If the discriminant is equal to zero, there is exactly one real root, meaning the parabola touches the x-axis at a single point.
- If the discriminant is less than zero, there are no real roots; instead, there are two complex roots.
Identifying Real Roots
Finding real roots of a quadratic equation is the goal when solving it. Real roots are the x-values where the parabola crosses the x-axis. An equation can have 0, 1, or 2 real roots, depending on the discriminant's value. In our example:
- The discriminant was positive, leading to two real roots.
- Using the quadratic formula, we found the two real roots as \(x_1 = -1.5\) and \(x_2 = \frac{1}{4}\).
- These roots tell us exactly at what points the parabola represented by \(-8x^2 - 10x + 3 = 0\) intersects the x-axis.
