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.

Quadratic Equations: Solved Examples

These work through the main question types one at a time: checking whether an equation is quadratic, setting one up from a word problem, solving it, and reading the nature of its roots.

Is it actually quadratic?

An equation is quadratic only once it's simplified: expand both sides fully and collect like terms first, then check the degree. Two examples with different outcomes:

$(x+1)^2 = 2(x-3)$ expands to $x^2+2x+1=2x-6$, which simplifies to $x^2 + 7 = 0$, degree 2, so this is a quadratic equation.

$(x-2)(x+1) = (x-1)(x+3)$ expands to $x^2 - x - 2 = x^2 + 2x - 3$, and the $x^2$ terms cancel, leaving $-3x + 1 = 0$, only degree 1, so this is not a quadratic equation, despite looking like one before expanding.

Turning a word problem into a quadratic equation

The product of two consecutive positive integers is 306. Represent this as a quadratic equation.

Solution: Let the first integer be $x$, so the next consecutive integer is $x+1$. Their product is 306:

Solving by factorization

Solve $2x^2 + x - 6 = 0$.

Solution: Split the middle term into two parts whose product matches $2 \times (-6) = -12$ and whose sum is $1$: that's $4$ and $-3$.

So $x = -2$ or $x = \dfrac{3}{2}$.

A word problem solved end to end

The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Solution: Let the base be $x$ cm, so the altitude is $x - 7$ cm. By the Pythagorean theorem:

A side length can't be negative, so $x = 12$: the base is 12 cm and the altitude is $12 - 7 = 5$ cm, a 5-12-13 right triangle, and $5^2 + 12^2 = 13^2$ checks out.

Solving by completing the square

Solve $2x^2 - 7x + 3 = 0$ by completing the square.

Solution: Divide by 2, then move the constant to the right:

Add $\left(\dfrac{7}{4}\right)^2 = \dfrac{49}{16}$ to both sides to complete the square on the left:

So $x = \dfrac{12}{4} = 3$ or $x = \dfrac{2}{4} = \dfrac{1}{2}$.

The nature of the roots

The discriminant $b^2-4ac$ tells you what kind of roots to expect before you even solve: positive means two distinct real roots, zero means one repeated real root, and negative means no real roots at all.

For what value(s) of $k$ does $2x^2+kx+3=0$ have two equal roots?

Solution: Equal roots require $b^2-4ac=0$: