Quadratic Equations and Inequalities: Methods, Applications, and Graphical Solutions

Quadratic Equations

Solving Quadratic Equations by Square Root Property

Quadratic equations can sometimes be solved using the square root property, which is applicable when the equation is in the form $(ax+b)^2 = c$.

• Square Root Property: If $(ax+b)^2 = c$, then $ax+b = \pm \sqrt{c}$.

• Steps:

  1. Isolate the squared term.

  2. Take the square root of both sides, remembering to include both positive and negative roots.

  3. Solve for the variable.

• Example: Solve $x^2 = 25$.

  • $x = \pm \sqrt{25}$

  • $x = \pm 5$

• Example: Solve $(x+2)^2 = 49$.

  • $x+2 = \pm \sqrt{49}$

  • $x+2 = \pm 7$

  • $x = 7-2 = 5$ or $x = -7-2 = -9$

Solving Quadratic Equations Using the Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation of the form $ax^2 + bx + c = 0$.

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

• Steps:

  1. Identify coefficients $a$, $b$, and $c$.

  2. Substitute into the formula.

  3. Simplify under the square root (discriminant).

  4. Calculate both possible values for $x$.

• Example: Solve $x^2 - 2x - 8 = 0$.

  • $a = 1$, $b = -2$, $c = -8$

  • $x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2}$

  • $x = \frac{2 \pm \sqrt{4 + 32}}{2}$

  • $x = \frac{2 \pm \sqrt{36}}{2}$

  • $x = \frac{2 \pm 6}{2}$

  • $x = 4$ or $x = -2$

Choosing a Method for Solving Quadratic Equations

There are several methods for solving quadratic equations, and the choice depends on the equation's form.

• Factoring: Used when the equation can be factored easily.

  • Example: $0.6x^2 - 0.6x - 5 = 0$

• Square Root Property: Used when the equation is in the form $(ax+b)^2 = c$.

• Quadratic Formula: Used when factoring is not possible or practical.

Applications of Quadratic Equations

Projectile Motion and Height Equations

Quadratic equations are used to model the height of objects in projectile motion, such as a diver jumping from a platform.

• General Formula: $h = \frac{1}{2}gt^2 + v_0t + h_0$

  • $g$ = acceleration due to gravity

  • $v_0$ = initial velocity

  • $h_0$ = initial height

• Example: A platform diver jumps from a platform that is 10 meters above the water.

  • Write the equation for height: $h = -4.9t^2 + 10$

  • Find the time to reach the water: Set $h = 0$ and solve $-4.9t^2 + 10 = 0$

  • $t = \sqrt{\frac{10}{4.9}} \approx 1.43$ seconds

Quadratic Inequalities

Solving Quadratic Inequalities

Quadratic inequalities involve expressions like $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$. Solutions are often represented graphically and in interval notation.

• Graphical Representation: The graph of $y = ax^2 + bx + c$ is a parabola. The solution to the inequality corresponds to the intervals where the parabola is above or below the x-axis.

• Example: Solve $x^2 - 4x + 3 < 0$.

  • Find roots: $x = 1$, $x = 3$

  • Solution: $1 < x < 3$

• Interval Notation: Solutions are written as intervals, e.g., $(1, 3)$.

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	Equation can be factored easily	$x^2 - 5x + 6 = 0$
Square Root Property	Equation is in the form $(ax+b)^2 = c$	$(x+2)^2 = 49$
Quadratic Formula	Any quadratic equation	$x^2 - 2x - 8 = 0$

Additional info: The notes also include applications to physics and interval notation for inequalities, which are standard in Intermediate Algebra.