10.4 Solve Applications Modeled by Quadratic Equations

Quadratic equations are powerful tools for solving real-world problems. They help us model situations involving area, motion, and more. By translating problem details into equations, we can find solutions that make sense in context.

Interpreting quadratic solutions requires careful consideration of what's realistic. We must relate mathematical answers back to the original problem, ensuring they fit within practical constraints. This process connects abstract math to concrete scenarios.

Solving Real-World Problems with Quadratic Equations

Applications of quadratic formula

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• Utilize the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to solve equations in the form $ax^2+bx+c=0$ where $a$, $b$, and $c$ are coefficients derived from the problem context
• Solve area problems by equating the area formula to a given value and solve for the unknown dimension (rectangle with length 3 units more than width and area of 70 square units, set up equation $x(x+3)=70$ and solve for width $x$)
• Solve consecutive integer problems by expressing the product of consecutive integers as a quadratic equation (product of two consecutive integers is 72, set up equation $x(x+1)=72$ and solve for smaller integer $x$)
• Solve projectile motion problems using the equation $h(t)=-16t^2+v_0t+h_0$ where $h(t)$ is height, $t$ is time, $v_0$ is initial velocity, and $h_0$ is initial height, solve for $t$ when given maximum height or time to reach specific height
  • The maximum height occurs at the vertex of the parabola formed by the projectile's path

Modeling with quadratic equations

• Identify unknown quantity and assign it a variable
• Translate given information into mathematical expressions
  • Represent relationships between quantities using equations ("the square of a number decreased by 5" translates to $x^2-5$)
• Construct quadratic equation by combining expressions according to problem context
  • Set up equation that equates two quantities or expresses a quantity as a specific value (square of a number decreased by 5 equals 20, equation is $x^2-5=20$)

Interpretation of quadratic solutions

• Relate solutions (roots) of quadratic equation back to problem context
  • Identify what variable represents and interpret meaning of solutions (solutions of 3 and -7 for width of rectangle, only 3 is valid since width cannot be negative)
• Assess reasonableness of solutions by checking if they make sense within problem context (solving for vehicle speed, negative value would be unreasonable)
• Round solutions to appropriate level of precision based on problem context, considering units and practical significance of digits (calculating room dimensions, round to nearest whole number or tenth of a unit)

Analyzing Quadratic Functions Graphically

• The graph of a quadratic function is a parabola
• The vertex represents the maximum or minimum point of the parabola
• The axis of symmetry is a vertical line passing through the vertex
• The discriminant ($b^2-4ac$) determines the number and nature of roots:
  • Positive: two distinct real roots
  • Zero: one repeated real root
  • Negative: no real roots