Quadratic Equations

What is a quadratic equation?

A quadratic equation is any equation of the form

where:

• $x$ represents a variable or an unknown
• $a$, $b$, and $c$ are constants
• $a \neq 0$

Here are some examples:

• $x^2 + 2x + 8 = 0$
• $4x^2 - 6x + 9 = 0$
• $x^2 - 25x = 0$
• $9x^2 + 81 = 0$

$9x + 5 = 0$ is not a quadratic equation since the $x^2$ term is missing: it's a linear equation.

A few more points to remember:

• A quadratic equation involves only one unknown ($x$ in this case).
• A quadratic equation only contains powers of $x$ that are non-negative.
• The solutions to the equation are the values of $x$ that make it equal 0. These are called the roots of the equation.

Solving by factoring

When a quadratic factors neatly, splitting the middle term is often faster than completing the square. Solve $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$.

Multiply the outer coefficients: $\sqrt{2} \times 5\sqrt{2} = 10$. Find two numbers that multiply to 10 and add to 7: that's 5 and 2. Split the middle term using them, then factor by grouping:

Each factor set to zero gives a root: $x = -\dfrac{5}{\sqrt{2}}$ or $x = -\sqrt{2}$.

Solving by completing the square

A quadratic equation with real or complex coefficients has two roots. These two roots may or may not be distinct, and they may or may not be real. One way to find them is a method called completing the square, starting from the standard form $ax^2 + bx + c = 0$:

1. Divide each side by $a$: $x^2 + \dfrac{bx}{a} + \dfrac{c}{a} = 0$
2. Subtract the constant $\dfrac{c}{a}$ from both sides: $x^2 + \dfrac{bx}{a} = -\dfrac{c}{a}$
3. Add the square of one-half of $\dfrac{b}{a}$, the coefficient of $x$, to both sides: $x^2 + \dfrac{bx}{a} + \dfrac{b^2}{4a^2} = -\dfrac{c}{a} + \dfrac{b^2}{4a^2}$
4. Rewrite the left side as a square and simplify the right side: $\left(x + \dfrac{b}{2a}\right)^2 = \dfrac{b^2 - 4ac}{4a^2}$
5. Take the square root of both sides: $x + \dfrac{b}{2a} = \pm\dfrac{\sqrt{b^2 - 4ac}}{2a}$, so $x = \pm\dfrac{\sqrt{b^2 - 4ac}}{2a} - \dfrac{b}{2a}$

This leads to the quadratic formula:

The $\pm$ means both $\dfrac{-b + \sqrt{b^2 - 4ac}}{2a}$ and $\dfrac{-b - \sqrt{b^2 - 4ac}}{2a}$ are roots of the equation.

Worked example

Let's solve $4x^2 - 12x + 3 = 0$ using this method.

1. Divide each side by 4: $x^2 - 3x + \dfrac{3}{4} = 0$
2. Subtract $\dfrac{3}{4}$ from both sides: $x^2 - 3x = -\dfrac{3}{4}$
3. Add the square of one-half of 3 (the coefficient of $x$) to both sides: $x^2 - 3x + \dfrac{9}{4} = -\dfrac{3}{4} + \dfrac{9}{4}$
4. Rewrite the left side as a square: $\left(x - \dfrac{3}{2}\right)^2 = \dfrac{6}{4}$
5. Take the square root of both sides: $x - \dfrac{3}{2} = \pm\dfrac{\sqrt{6}}{2}$, so $x = \dfrac{3 \pm \sqrt{6}}{2} \approx 2.72 \text{ or } 0.28$

The shape of the graph

The graph of a quadratic equation is a curve called a parabola. When $a$ is positive, the parabola opens upward; when $a$ is negative, it opens downward. As $|a|$ increases, the parabola gets narrower. (If $a = 0$, it's no longer a quadratic equation at all.) Changing $b$ shifts the vertex left or right of the y-axis without changing the parabola's shape, and changing $c$ shifts the whole graph up or down.

Quadratic equations in real life

Quadratic equations show up whenever a quantity depends on the square of another, such as finding the speed of a moving object, a product's profit, or an area. Here's one example: finding the speed of a boat that travels up and down a river.

A river flows at 2 km/hour. A boat travels 20 km upstream against the current and then back, and the round trip takes 4 hours. Let $x$ be the boat's speed in still water.

Going upstream, the boat's speed relative to the ground is $x - 2$ (the current slows it down); going downstream it's $x + 2$ (the current helps it along). Since $\text{time} = \dfrac{\text{distance}}{\text{speed}}$ and the total time is 4 hours:

Expanding this algebraically gives $x^2 - 10x - 4 = 0$. Solving with the quadratic formula gives two values: $x \approx 10.38$ and $x \approx -0.38$. The negative value has no physical meaning here, so the boat's speed in still water is about 10.38 km/hour.