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Magazine Chapter 10: Problem 58
Use the quadratic formula to solve each of the following equations. Express the solutions to the nearest hundredth. $$5 x^{2}-11 x-14=0$$
Short Answer
Expert verified
The solutions are \(x = 3.00\) and \(x = -0.90\).
Step by step solution
01
Identify Coefficients
The quadratic equation is in the form \(ax^2 + bx + c = 0\). For the equation \(5x^2 - 11x - 14 = 0\), identify the coefficients as follows: \(a = 5\), \(b = -11\), and \(c = -14\).
02
Write the Quadratic Formula
The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula will be used to find the roots of the quadratic equation.
03
Calculate the Discriminant
The discriminant of the quadratic equation is \(b^2 - 4ac\). Substitute the values \(b = -11\), \(a = 5\), and \(c = -14\) into the formula: \((-11)^2 - 4 \times 5 \times (-14) = 121 + 280 = 401\).
04
Compute the Square Root of the Discriminant
Calculate the square root of the discriminant: \(\sqrt{401} \approx 20.025\).
05
Solve for the Roots
Substitute \(b = -11\), \(\sqrt{401} = 20.025\), and \(a = 5\) into the quadratic formula to find the roots: \[x = \frac{11 \pm 20.025}{10}\]Solve separately for each case:1. \(x = \frac{11 + 20.025}{10} = 3.0025\)2. \(x = \frac{11 - 20.025}{10} = -0.9025\).
06
Round the Solutions
To express the solutions to the nearest hundredth, round the calculated roots:1. \(x = 3.0025\) rounds to \(x = 3.00\)2. \(x = -0.9025\) rounds to \(x = -0.90\).
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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 key part of the quadratic formula, an important tool for solving quadratic equations. It is found inside the square root in the quadratic formula, represented by the expression \(b^2 - 4ac\). This part of the formula informs us about the nature of the roots of the equation, even before we compute them.
Here's how the discriminant works:
Here's how the discriminant works:
- **Positive Discriminant:** If \(b^2 - 4ac > 0\), then the quadratic equation has two distinct real solutions. This is the case in our exercise where the discriminant is 401.
- **Zero Discriminant:** If \(b^2 - 4ac = 0\), there is exactly one real solution, which means the parabola just "touches" the x-axis at a single point.
- **Negative Discriminant:** If \(b^2 - 4ac < 0\), the equation does not have any real solutions; instead, it has two complex roots.
Quadratic Equation
A quadratic equation is a second-order polynomial equation in one variable, generally expressed as \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants where \(a eq 0\). Quadratic equations form parabolic graphs when plotted on the coordinate axis.
Let's break down the components:
Let's break down the components:
- **\(a\):** The coefficient of \(x^2\). It determines the direction and the width of the parabola. A positive \(a\) makes the parabola open upwards, while a negative \(a\) opens it downwards.
- **\(b\):** The coefficient of \(x\), influencing the parabola's position relative to the y-axis.
- **\(c\):** The constant term, which affects the vertical position of the parabola on the graph.
Solving Equations
Solving a quadratic equation like \(5x^2 - 11x - 14 = 0\) typically involves finding the values of \(x\) that satisfy the equation. The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is a reliable method for finding these solutions.
Here are the steps involved in solving quadratic equations using the quadratic formula:
Here are the steps involved in solving quadratic equations using the quadratic formula:
- **Identify Coefficients:** Start by identifying \(a\), \(b\), and \(c\) from the equation. These are the numbers you will substitute into the quadratic formula.
- **Calculate the Discriminant:** Compute the discriminant \(b^2 - 4ac\) to determine the nature of the roots.
- **Square Root of Discriminant:** Find the square root of the discriminant. This value will be added and subtracted from \(-b\) in the formula.
- **Solve for \(x\):** Substitute back into the formula, evaluate each case of the \(\pm\) to find the values of \(x\).
- **Round to Nearest Hundredth:** Once you have solutions, round them to the appropriate decimal places if required, as seen in our exercise where solutions are limited to two decimal points for precision.
