2.5 Quadratic Equations

Quadratic equations describe relationships where one variable is squared, taking the standard form $ax^2 + bx + c = 0$. They show up constantly in physics, economics, and engineering problems, so getting comfortable with the different solving methods here will pay off throughout the course.

You have several tools for solving these: factoring, the square root property, completing the square, and the quadratic formula. Each works best in different situations, and this guide covers when and how to use each one.

Solving Quadratic Equations

Factoring techniques for quadratics

Factoring is usually the fastest method when it works. The idea is to rewrite the quadratic as a product of two binomials, then use the Zero Product Property to find the solutions.

The Zero Product Property says: if $ab = 0$, then $a = 0$ or $b = 0$ (or both). This is what lets you break a factored equation into two simpler equations.

Steps for solving by factoring:

1. Write the equation in standard form: $ax^2 + bx + c = 0$
2. Factor the quadratic expression completely
3. Set each factor equal to zero
4. Solve each resulting equation

For example, with $x^2 - 5x + 6 = 0$:

• Factor: $(x - 2)(x - 3) = 0$
• Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$
• Solutions: $x = 2$ or $x = 3$

These solutions are also called the roots of the equation.

Special factoring patterns to recognize:

• Difference of squares: $a^2 - b^2 = (a + b)(a - b)$
  • Example: $x^2 - 9 = (x + 3)(x - 3)$
• Perfect square trinomials: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$
  • Example: $x^2 + 6x + 9 = (x + 3)^2$

Factoring by grouping is useful when the leading coefficient $a \neq 1$. You split the middle term, group in pairs, and factor out the GCF from each group. For instance, $6x^2 + 3x$ can be factored as $3x(2x + 1)$ by pulling out the GCF.

Square root property in equations

The square root property is the go-to method when your equation has a squared expression but no loose $x$ term (no $bx$). It states:

The $\pm$ is critical. Since both positive and negative numbers give a positive result when squared, you always need both solutions.

Steps:

1. Isolate the squared term on one side of the equation
2. Take the square root of both sides
3. Include $\pm$ on the result
4. Simplify if possible

For example, with $x^2 + 4 = 20$:

• Isolate: $x^2 = 16$
• Take the square root: $x = \pm\sqrt{16}$
• Simplify: $x = \pm 4$

A common mistake is forgetting the negative solution. Always write $\pm$.

Completing the square method

Completing the square works on any quadratic, even when factoring doesn't. It transforms the equation into a perfect square trinomial so you can apply the square root property.

Steps:

1. Move the constant to the other side: $x^2 + bx = -c$
2. Take half the coefficient of $x$, then square it: $\left(\frac{b}{2}\right)^2$
3. Add that value to both sides of the equation
4. Factor the left side as a perfect square: $\left(x + \frac{b}{2}\right)^2$
5. Apply the square root property and solve for $x$

For example, with $x^2 + 6x + 5 = 0$:

• Move the constant: $x^2 + 6x = -5$
• Half of 6 is 3, and $3^2 = 9$. Add 9 to both sides: $x^2 + 6x + 9 = 4$
• Factor: $(x + 3)^2 = 4$
• Square root property: $x + 3 = \pm 2$
• Solutions: $x = -3 + 2 = -1$ or $x = -3 - 2 = -5$

Completing the square also converts a quadratic into vertex form: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola and $x = h$ is the axis of symmetry. This is useful for graphing because:

• The parabola opens upward if $a > 0$ and downward if $a < 0$
• The graph is symmetric about the axis of symmetry
• You can plot the vertex first, then find additional points using a table of values

Quadratic Formula and Applications

Quadratic formula applications

The quadratic formula works on every quadratic equation, no matter what. When factoring isn't obvious or completing the square gets messy, this is your reliable fallback:

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

To use it, identify $a$, $b$, and $c$ from the standard form $ax^2 + bx + c = 0$, plug them in, and simplify. Be careful with signs, especially when $b$ or $c$ is negative.

The discriminant $\Delta = b^2 - 4ac$ tells you what kind of solutions to expect before you even solve:

• $\Delta > 0$: Two distinct real solutions. Example: $x^2 - 5x + 6 = 0$ has $\Delta = 25 - 24 = 1$, giving $x = 2$ or $x = 3$
• $\Delta = 0$: One repeated real solution. Example: $x^2 - 6x + 9 = 0$ has $\Delta = 36 - 36 = 0$, giving $x = 3$
• $\Delta < 0$: No real solutions (only complex solutions). Example: $x^2 + 2x + 5 = 0$ has $\Delta = 4 - 20 = -16$

After plugging into the formula, always simplify your answer by reducing square roots and fractions.

Word problem applications frequently use quadratics. For projectile motion, the height of an object is often modeled by something like $h(t) = -16t^2 + 64t + 5$, where $t$ is time in seconds. Setting $h(t) = 0$ and solving tells you when the object hits the ground. Always check whether both solutions make sense in context (negative time, for instance, usually doesn't).

Additional Concepts in Quadratic Equations

• A quadratic equation is a polynomial with a highest exponent of 2. The values $a$, $b$, and $c$ are called the coefficients.
• Graphing a quadratic produces a parabola, a U-shaped curve that opens up or down.
• The solutions (roots) of a quadratic correspond to the x-intercepts of the parabola, the points where the graph crosses the x-axis. If the discriminant is negative, the parabola doesn't cross the x-axis at all.