Lesson 6: Quadratic Equations

New Concept

We're leveling up from linear to quadratic equations, where the variable is squared ($ax^2 + bx + c = 0$). You'll learn how to solve these by factoring and using the powerful Zero Product Property to tackle new types of problems.

What’s next

Now, let's master this concept. Next up are worked examples and interactive practice cards to build your factoring and solving skills.

Property

An equation of the form $ax^2 + bx + c = 0$ is called a quadratic equation.

Examples

• The equation $x^2 - 5x + 6 = 0$ is a quadratic equation in standard form with $a=1$, $b=-5$, and $c=6$.
• The equation $4y^2 = 12$ is a quadratic equation. We can write it in standard form as $4y^2 - 0y - 12 = 0$.
• The equation $k(k-3) = 10$ is also a quadratic equation. Expanding it gives $k^2 - 3k = 10$, which becomes $k^2 - 3k - 10 = 0$ in standard form.

Explanation

A quadratic equation is a polynomial equation where the variable's highest power is two (it's squared). Unlike linear equations, they often have two solutions. The condition $a \neq 0$ is crucial, otherwise, it's not a quadratic equation.

Property

If $a \cdot b = 0$, then either $a = 0$ or $b = 0$ or both.

Examples

• To solve $(x-3)(x+5) = 0$, we set each factor to zero: $x-3=0$ or $x+5=0$. The solutions are $x=3$ and $x=-5$.
• For $7y(2y-1)=0$, we set $7y=0$ or $2y-1=0$. This gives the solutions $y=0$ and $y=\frac{1}{2}$.
• If $(z+4)^2 = 0$, it means $(z+4)(z+4)=0$. Both factors give the same solution, $z=-4$. This is called a double root.

Explanation

This property is the secret to solving factored equations. If a product of several things equals zero, at least one of those things must be zero. This lets us break a complicated product into simpler, separate equations.

Property

To use the Zero Product Property, the quadratic equation must be factored, with zero on one side. We must start with the quadratic equation in standard form, $ax^2 + bx + c = 0$.
HOW TO Solve a Quadratic Equation by Factoring.
Step 1. Write the quadratic equation in standard form, $ax^2 + bx + c = 0$.
Step 2. Factor the quadratic expression.
Step 3. Use the Zero Product Property.
Step 4. Solve the linear equations.
Step 5. Check.

Examples

• To solve $x^2 - x - 12 = 0$, we first factor it into $(x-4)(x+3)=0$. Using the Zero Product Property, we get $x=4$ and $x=-3$.
• To solve $3y^2 - 7y = -2$, first write it in standard form: $3y^2 - 7y + 2 = 0$. Factoring gives $(3y-1)(y-2)=0$, so the solutions are $y=\frac{1}{3}$ and $y=2$.
• To solve $k^2 = 8k$, rewrite it as $k^2 - 8k = 0$. Factor out the common factor $k(k-8)=0$. The solutions are $k=0$ and $k=8$.

Explanation

Factoring transforms a single, complex quadratic problem into a product of simple linear factors that equals zero. This allows you to apply the Zero Product Property and solve for the variable by handling each factor separately.

Property

Use a problem-solving strategy for applications:
Step 1. Read the problem to understand it.
Step 2. Identify what you are looking for.
Step 3. Name what you are looking for with a variable.
Step 4. Translate the problem into an equation. For right triangles, the Pythagorean Theorem $a^2 + b^2 = c^2$ is often used.
Step 5. Solve the equation.
Step 6. Check if the answer makes sense in the context of the problem.
Step 7. Answer the question with a complete sentence.

Examples

• The product of two consecutive odd integers is 99. Find the integers. Let the integers be $n$ and $n+2$. The equation is $n(n+2)=99$. This gives $n^2+2n-99=0$, which factors to $(n+11)(n-9)=0$. The pairs are $-11, -9$ and $9, 11$.
• A rectangular garden has an area of 54 square feet. Its length is 3 feet more than its width. Find its dimensions. Let the width be $w$. The length is $w+3$. The equation is $w(w+3)=54$, or $w^2+3w-54=0$. Factoring gives $(w+9)(w-6)=0$. Since width cannot be negative, the width is 6 feet and the length is 9 feet.
• A right triangle has a hypotenuse of 20 inches. One leg is 4 inches longer than the other. Find the leg lengths. Let the legs be $x$ and $x+4$. By the Pythagorean Theorem, $x^2 + (x+4)^2 = 20^2$. This simplifies to $2x^2+8x-384=0$, or $x^2+4x-192=0$. Factoring gives $(x+16)(x-12)=0$. The legs are 12 and 16 inches.

Explanation

Quadratic equations are powerful tools for modeling real-world scenarios involving area, consecutive numbers, or geometric shapes like right triangles. Translating a word problem into a mathematical equation allows you to find a precise solution.