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Magazine Chapter 6: Problem 76
$$ \text { For Problems 51-80, solve each equation. (Objective 2) } $$ $$ 2 x^{2}+7 x-30=0 $$
Short Answer
Expert verified
x = 2.5 and x = -6
Step by step solution
01
Identify the Quadratic Equation
The given equation is a quadratic equation since it is in the form \( ax^2 + bx + c = 0 \). Here, \( a = 2 \), \( b = 7 \), and \( c = -30 \).
02
Apply the Quadratic Formula
To solve the quadratic equation \( ax^2 + bx + c = 0 \), use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the values of \( a \), \( b \), and \( c \) into the formula.
03
Calculate the Discriminant
Compute the discriminant \( b^2 - 4ac \): \( 7^2 - 4(2)(-30) = 49 + 240 = 289 \). The positive discriminant indicates real and distinct solutions.
04
Simplify Under the Square Root
Since the discriminant is \( 289 \), calculate its square root: \( \sqrt{289} = 17 \).
05
Solve for x
Substitute back into the quadratic formula: \( x = \frac{-7 \pm 17}{4} \). This results in two solutions.
06
Calculate First Solution
Solve \( x = \frac{-7 + 17}{4} = \frac{10}{4} = 2.5 \).
07
Calculate Second Solution
Solve \( x = \frac{-7 - 17}{4} = \frac{-24}{4} = -6 \).
08
State the Solutions
The solutions to the quadratic equation \( 2x^2 + 7x - 30 = 0 \) are \( x = 2.5 \) and \( x = -6 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Discriminant
In a quadratic equation, the discriminant is a key component for determining the nature of the solutions. It is part of the quadratic formula and is represented as \( b^2 - 4ac \).
Here's how it works:
Here's how it works:
- If the discriminant is greater than zero, \( b^2 - 4ac > 0 \), the equation has two distinct real solutions.
- If the discriminant equals zero, \( b^2 - 4ac = 0 \), there is exactly one real solution, also known as a repeated or double root.
- If the discriminant is less than zero, \( b^2 - 4ac < 0 \), there are no real solutions. Instead, the solutions are complex or imaginary.
Quadratic Formula
The quadratic formula is a foolproof method for solving quadratic equations. It is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This versatile formula can solve any quadratic equation of the form \( ax^2 + bx + c = 0 \) without the need to factorize or complete the square. Let's break down the process:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This versatile formula can solve any quadratic equation of the form \( ax^2 + bx + c = 0 \) without the need to factorize or complete the square. Let's break down the process:
- Identify \( a \), \( b \), and \( c \) from the equation.
- Calculate the discriminant \( b^2 - 4ac \).
- Substitute these values into the quadratic formula.
- Simplify to find the potential solutions for \( x \).
Real Solutions
Real solutions to quadratic equations are values of \( x \) that satisfy the equation and are not imaginary. To find real solutions, one must first confirm that the discriminant is non-negative (zero or positive).
In the quadratic formula, real solutions occur when:
In the quadratic formula, real solutions occur when:
- The discriminant \( b^2 - 4ac \) is positive, leading to two different numbers for \( x \).
- The discriminant is zero, resulting in one repeated real solution.
