Textbook Question
Solve each equation in Exercises 65–74 using the quadratic formula.
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1. Equations & Inequalities
Intro to Quadratic Equations
3:28 minutesProblem 72
Textbook Question
Without solving the given quadratic equation, determine the number and type of solutions.
Verified step by step guidance1
Rewrite the given quadratic equation in standard form \(ax^2 + bx + c = 0\). Start by moving all terms to one side: \$9x^2 + 3x - 2 = 0$.
Identify the coefficients: \(a = 9\), \(b = 3\), and \(c = -2\).
Calculate the discriminant using the formula \(\Delta = b^2 - 4ac\).
Substitute the values into the discriminant formula: \(\Delta = (3)^2 - 4(9)(-2)\).
Analyze the discriminant value: if \(\Delta > 0\), there are two distinct real solutions; if \(\Delta = 0\), there is one real repeated solution; if \(\Delta < 0\), there are two complex solutions.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Quadratic Equation
A quadratic equation is typically written in the form ax² + bx + c = 0, where a, b, and c are constants. Converting the given equation into this form is essential for analyzing its properties, such as the number and type of solutions.
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Discriminant of a Quadratic Equation
The discriminant, given by Δ = b² - 4ac, determines the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real solutions; if Δ = 0, there is one real repeated solution; and if Δ < 0, there are two complex conjugate solutions.
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Types of Solutions for Quadratic Equations
Quadratic equations can have real or complex solutions. Real solutions occur when the graph of the quadratic intersects the x-axis, while complex solutions occur when it does not. Understanding this helps in interpreting the discriminant without solving the equation.
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