10.4 Solve Applications Modeled by Quadratic Equations

Solving Real-World Problems with Quadratic Equations

Quadratic equations show up whenever a real-world problem involves a squared quantity. That includes area calculations, projectile motion, and number relationships. The key skill here is translating a word problem into an equation, solving it, and then checking that your answer actually makes sense in context.

Modeling with quadratic equations

Before you can solve anything, you need to turn the word problem into math. Here's a reliable process:

1. Read the problem carefully and identify the unknown quantity. Assign it a variable (usually $x$).

2. Translate relationships into expressions. For example, "the square of a number decreased by 5" becomes $x^2 - 5$.

3. Build the equation by setting expressions equal to each other based on what the problem tells you. If "the square of a number decreased by 5 equals 20," your equation is $x^2 - 5 = 20$.

4. Rearrange into standard form $ax^2 + bx + c = 0$ so you can solve using factoring, the square root property, or the quadratic formula.

The hardest part is usually step 2. Pay close attention to phrases like "more than," "less than," "product of," and "decreased by." These tell you which operations to use.

Applications of the quadratic formula

The quadratic formula solves any equation in the form $ax^2 + bx + c = 0$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here $a$, $b$, and $c$ are the coefficients from your equation. You'll use this formula (or factoring, when it's simpler) across several common problem types.

Area problems: These involve a shape whose dimensions are described in terms of one variable.

Example: A rectangle's length is 3 units more than its width, and its area is 70 square units. If the width is $x$, then the length is $x + 3$, so:

$x(x + 3) = 70$ $x^2 + 3x - 70 = 0$

Factoring gives $(x + 10)(x - 7) = 0$, so $x = -10$ or $x = 7$. Since width can't be negative, the width is 7 units and the length is 10 units.

Consecutive integer problems: Consecutive integers can be written as $x$ and $x + 1$ (or $x$ and $x + 2$ for consecutive even or odd integers).

Example: The product of two consecutive integers is 72. Set up $x(x + 1) = 72$, which gives $x^2 + x - 72 = 0$. Factoring: $(x + 9)(x - 8) = 0$, so $x = -9$ or $x = 8$. Both solutions are valid here: the integers are either 8 and 9, or $-9$ and $-8$.

Projectile motion problems: Objects launched or thrown follow the model:

$h(t) = -16t^2 + v_0 t + h_0$

where $h(t)$ is the height in feet at time $t$ seconds, $v_0$ is the initial velocity (ft/s), and $h_0$ is the starting height. To find when the object reaches a certain height, set $h(t)$ equal to that height and solve for $t$. The maximum height occurs at the vertex of the parabola, which is at $t = \frac{-v_0}{2(-16)} = \frac{v_0}{32}$.

Interpretation of quadratic solutions

The quadratic formula often gives you two solutions. Your job is to figure out which ones (if not both) make sense for the problem.

• Check against physical constraints. A length, width, or speed can't be negative. Time is usually non-negative. If one solution violates these constraints, discard it.
• Both solutions can sometimes be valid. In the consecutive integer example above, both $x = 8$ and $x = -9$ produce legitimate pairs of consecutive integers.
• Round appropriately. If you're calculating room dimensions, rounding to the nearest tenth of a foot is reasonable. If you're counting people or objects, round to a whole number. Always include units in your final answer.

A common mistake is solving the equation correctly but forgetting to answer the actual question. If the problem asks for the length and you solved for the width, you still need one more step.

Analyzing Quadratic Functions Graphically

The graph of a quadratic function is a parabola. A few features connect directly to the application problems above:

• The vertex is the highest or lowest point on the parabola. In projectile problems, this represents the maximum height.
• The axis of symmetry is the vertical line through the vertex, at $x = \frac{-b}{2a}$.
• The discriminant $b^2 - 4ac$ tells you how many real solutions exist:
  • Positive: two distinct real roots (the parabola crosses the x-axis twice)
  • Zero: one repeated real root (the parabola touches the x-axis at its vertex)
  • Negative: no real roots (the parabola doesn't cross the x-axis, meaning the situation described by the equation has no real solution)