Find the number of solutions

Learning Objectives Define and identify the three possible outcomes for the number of solutions to a linear equation (one, none, infinite). Analyze a linear equation without fully solving it to determine the number of solutions. Calculate the discriminant of a quadratic equation in the form ax² + bx + c = 0. Use the discriminant to determine the number of real solutions (two, one, or zero) for a quadratic equation. Connect the number of solutions of a quadratic equation to the number of x-intercepts on the graph of its corresponding parabola. Distinguish between finding the actual solutions and finding *how many* solutions exist. Ever tried to solve a puzzle and wondered if there was more than one right answer, or maybe none at all? 🤔 Equations are just like that! In this...

TermDefinitionExample Solution (or Root)A value for a variable that makes an equation a true statement.For the equation x + 3 = 5, the solution is x = 2 because 2 + 3 = 5. Linear EquationAn equation where the highest power of the variable is 1. Its graph is a straight line.3x - 7 = 2 Quadratic EquationAn equation where the highest power of the variable is 2, written in the standard form ax² + bx + c = 0, where a ≠ 0.x² - 4x + 3 = 0 DiscriminantThe expression b² - 4ac from the quadratic formula. Its value 'discriminates' or determines the number of real solutions a quadratic equation has.For 2x² + 5x - 3 = 0, the discriminant is 5² - 4(2)(-3) = 49. IdentityAn equation that is true for all possible values of the variable. It has infinitely many solutions.2(x + 1) = 2x + 2 simplifi...

Number of Solutions for Linear Equations After simplifying a linear equation to the form ax = b: 1. One solution if a ≠ 0. 2. No solution if a = 0 and b ≠ 0. 3. Infinite solutions if a = 0 and b = 0. Use this after simplifying a linear equation to quickly determine the number of solutions without finding the exact value. It helps classify the equation as having a unique solution, being a contradiction, or being an identity. The Discriminant Formula Δ = b² - 4ac This formula is used for quadratic equations in standard form (ax² + bx + c = 0). You must identify the coefficients a, b, and c before substituting them into the formula to find the value of the discriminant (Δ). Interpreting the Discriminant For a quadratic equation: 1. If Δ > 0, there are two distinct...