Quadratic Equations — Core Skills

Quadratic Equations: The Basics

Part 1 of 2 — One Skill, One Idea

A quadratic is an equation where the highest power of $x$ is $2$. Something like $x^{2} + 7x + 12 = 0$.

To factor means to rewrite a sum as a multiplication. For example, $12$ factored is $3 \times 4$. With quadratics, you rewrite $x^{2} + 7x + 12$ as two parentheses multiplied together.

A solution (also called a zero or a root) is a value of $x$ that makes the equation true.

The one move: find two numbers

For $x^{2} + bx + c$, look for two numbers that multiply to $c$ and add to $b$. Those two numbers go inside the parentheses.

Worked example

Solve $x^{2} + 7x + 12 = 0$.

Step 1 — Name the parts. Here $b = 7$ and $c = 12$.

Step 2 — Find two numbers that multiply to $12$ and add to $7$. Try the pairs that multiply to $12$: $1$ and $12$ add to $13$. $2$ and $6$ add to $8$. $3$ and $4$ add to $7$. That is the pair.

Step 3 — Write the factored form: $(x + 3)(x + 4) = 0$.

Step 4 — Use the zero rule. If two things multiply to zero, at least one of them must be zero. So either $x + 3 = 0$ or $x + 4 = 0$.

Step 5 — Solve each small equation. From $x + 3 = 0$, subtract $3$ from both sides: $x = -3$. From $x + 4 = 0$, subtract $4$: $x = -4$.

The solutions are $x = -3$ and $x = -4$.

The sign check

Notice the signs flip. A factor of $(x + 3)$ gives the solution $x = -3$. A factor of $(x - 3)$ gives the solution $x = 3$. Set each parenthesis equal to zero and solve, every time, and the signs take care of themselves.

Quadratic Equations: Practice

Part 2 of 2 — Run the Steps

To solve $x^{2} + bx + c = 0$ by factoring:

1. Make sure one side is $0$. Move everything to the left if you have to.
2. Find two numbers that multiply to $c$ and add to $b$.
3. Write $(x + \text{first number})(x + \text{second number}) = 0$.
4. Set each parenthesis equal to $0$ and solve the two small equations.
5. Write both answers.

Two shortcuts you will use often

No constant term. If the equation looks like $x^{2} - 5x = 0$, pull out the shared $x$: $x(x - 5) = 0$. Then $x = 0$ or $x = 5$. Do not divide both sides by $x$ — that would throw away the answer $x = 0$.

A plain square. If the equation looks like $x^{2} = 36$, take the square root of both sides and keep both signs: $x = 6$ or $x = -6$. Both work, because $6 \times 6 = 36$ and $(-6) \times (-6) = 36$.

A quick sign guide

• Both numbers positive when $b$ and $c$ are both positive.
• Both numbers negative when $c$ is positive and $b$ is negative.
• One positive and one negative when $c$ is negative.