Quadratic Equations: Methods, Properties, and Applications

Quadratic Equations

Definition of a Quadratic Equation

A quadratic equation in x is an equation that can be written in the standard form:

$ax^2 + bx + c = 0$

• a, b, and c are real numbers, with a ≠ 0.

• Also called a second-degree polynomial equation in x.

Solving Quadratic Equations

Solving by Factoring

Factoring is an efficient method when the quadratic can be easily factored into two binomials.

• Step 1: Rewrite the equation in standard form, moving all nonzero terms to one side to obtain zero on the other side.

• Step 2: Factor the quadratic expression completely.

• Step 3: Apply the Zero-Product Principle: If the product of two expressions is zero, at least one factor must be zero.

• Step 4: Set each factor containing a variable equal to zero and solve the resulting equations.

• Step 5: Check the solutions in the original equation.

Example: Solve $x^2 - 5x + 6 = 0$ by factoring.

• Factor: $(x - 2)(x - 3) = 0$

• Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$

• Solutions: $x = 2$ or $x = 3$

Solving by the Square Root Property

Quadratic equations of the form $u^2 = d$ can be solved using the Square Root Property.

• If $u^2 = d$, then $u = \pm \sqrt{d}$

• Equivalently, $x = a \pm b$ if $(x - a)^2 = b^2$

Example: Solve $(x - 4)^2 = 9$.

• Take square roots: $x - 4 = \pm 3$

• So $x = 4 + 3 = 7$ or $x = 4 - 3 = 1$

Solving by Completing the Square

Completing the square is a method that rewrites a quadratic in the form $(x + p)^2 = q$.

• Given $x^2 + bx = c$, add $\left(\frac{b}{2}\right)^2$ to both sides to form a perfect square trinomial.

• Result: $x^2 + bx + \left(\frac{b}{2}\right)^2 = c + \left(\frac{b}{2}\right)^2$

• Then $(x + \frac{b}{2})^2 = c + \left(\frac{b}{2}\right)^2$

• Solve for $x$ by taking square roots.

Example: Solve $x^2 + 6x + 5 = 0$ by completing the square.

• Move constant: $x^2 + 6x = -5$

• Add $(\frac{6}{2})^2 = 9$: $x^2 + 6x + 9 = 4$

• So $(x + 3)^2 = 4$

• Take square roots: $x + 3 = \pm 2$

• Solutions: $x = -3 + 2 = -1$ or $x = -3 - 2 = -5$

Solving by the Quadratic Formula

The Quadratic Formula provides solutions for any quadratic equation in standard form.

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

• Identify coefficients a, b, and c from $ax^2 + bx + c = 0$.

• Substitute into the formula to find the solutions.

Example: Solve $2x^2 - 4x - 6 = 0$.

• Here, $a = 2$, $b = -4$, $c = -6$

• Plug into the formula:

$x = \frac{ -(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)} }{2 \cdot 2}$ $x = \frac{ 4 \pm \sqrt{16 + 48} }{4}$ $x = \frac{ 4 \pm \sqrt{64} }{4}$ $x = \frac{ 4 \pm 8 }{4}$ $x = 3$ or $x = -1$

The Discriminant

The discriminant is the expression under the square root in the quadratic formula:

$D = b^2 - 4ac$

• The value of D determines the number and type of solutions:

Discriminant ($D$)	Nature of Solutions
$D > 0$	Two distinct real solutions
$D = 0$	One real (repeated) solution
$D < 0$	Two complex (imaginary) solutions

Example: For $x^2 - 6x + 9 = 0$, $D = (-6)^2 - 4 \cdot 1 \cdot 9 = 36 - 36 = 0$ (one real solution).

Choosing the Most Efficient Method

• Factoring: Use when the quadratic factors easily.

• Square Root Property: Use for equations of the form $(x + p)^2 = q$.

• Completing the Square: Useful when the coefficient of $x^2$ is 1 and $b$ is even.

• Quadratic Formula: Works for all quadratics, especially when factoring is difficult.

Applications of Quadratic Equations

Modeling with Quadratic Equations

Quadratic equations are used to model various real-world situations, such as projectile motion, area problems, and physiological formulas.

Example: The formula $P = 0.5A + 90$ models a woman’s normal systolic blood pressure, P, at age A. To find the age for $P = 115$, solve $115 = 0.5A + 90$:

• Subtract 90: $25 = 0.5A$

• Divide by 0.5: $A = 50$

(If the equation is quadratic, use appropriate methods as above.)

The Pythagorean Theorem

The Pythagorean Theorem relates the sides of a right triangle:

$a^2 + b^2 = c^2$

• a and b are the lengths of the legs.

• c is the length of the hypotenuse.

Example: If a TV screen has width 24 inches and height 18 inches, the diagonal (screen size) is:

$c = \sqrt{24^2 + 18^2} = \sqrt{576 + 324} = \sqrt{900} = 30$ inches

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example Equation
Factoring	Equation factors easily	$x^2 - 5x + 6 = 0$
Square Root Property	Equation in form $(x + p)^2 = q$	$(x - 3)^2 = 16$
Completing the Square	Coefficient of $x^2$ is 1, $b$ is even	$x^2 + 6x + 5 = 0$
Quadratic Formula	Any quadratic equation	$2x^2 - 4x - 6 = 0$

Additional info: The above notes include expanded explanations and examples for each method, as well as a summary table for quick reference.