Quadratic Equations Worksheets

What is a quadratic equation?

A quadratic equation is an equation in which the highest power of the variable is 2. In standard form it is written $ax^{2} + bx + c = 0$, where $a \neq 0$. Because of the squared term, a quadratic can have two different solutions, and students usually meet it in grades 9–11, right after they learn to factor trinomials.

Every equation on these worksheets can be solved by factoring. Factoring turns the equation into a product that equals zero, and then the zero-product property finishes the job: if $A \cdot B = 0$, then $A = 0$ or $B = 0$. Each factor gives one solution.

The three difficulty levels build on each other:

• Easy — the leading coefficient is 1, as in $x^{2} + bx + c = 0$, and both solutions are integers.
• Medium — the leading coefficient is 2, 3 or 4, so one solution is usually a fraction.
• Hard — the equation is given as $x^{2} + bx = k$ and must be put into standard form before it can be factored.

Worked examples

Easy: $x^{2} - x - 12 = 0$

Look for two numbers that multiply to $-12$ and add to $-1$. The pair $-4$ and $3$ works.

Set each factor equal to zero: $x - 4 = 0$ gives $x = 4$, and $x + 3 = 0$ gives $x = -3$. Answer: $x = -3,\ 4$.

Medium: $2x^{2} + x - 15 = 0$

Multiply $a \cdot c = 2 \cdot (-15) = -30$. Find two numbers that multiply to $-30$ and add to $b = 1$: they are $6$ and $-5$. Split the middle term and factor by grouping:

$x + 3 = 0$ gives $x = -3$, and $2x - 5 = 0$ gives $x = \tfrac{5}{2}$. Answer: $x = -3,\ \tfrac{5}{2}$.

Hard: $x^{2} + 2x = 24$

The right side is not zero, so the zero-product property cannot be used yet. Subtract 24 from both sides first:

Two numbers that multiply to $-24$ and add to $2$ are $6$ and $-4$, so $(x + 6)(x - 4) = 0$. Answer: $x = -6,\ 4$.

Check your answers

Substitute each solution back into the original equation. For the hard example, $x = 4$ gives $16 + 8 = 24$ and $x = -6$ gives $36 - 12 = 24$, so both solutions are correct. Checking takes a few seconds and catches almost every sign error.

Common mistakes

• Factoring before the equation equals zero. Writing $x(x + 2) = 24$ and concluding $x = 24$ is wrong: a product equal to 24 says nothing about either factor on its own. Only zero has that property.
• Reading the sign backwards. The factor $x + 3$ gives the solution $x = -3$, not $x = 3$. Solve each factor as its own small equation instead of copying the number.
• Forgetting the leading coefficient. From $2x - 5 = 0$ the solution is $x = \tfrac{5}{2}$, not $x = 5$.
• Losing a solution. A quadratic that factors into two different factors has two solutions. Stopping after the first one is the most common way to lose marks.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

1. $x^{2} + 8x + 7 = 0$
2. $x^{2} - 1 = 0$
3. $x^{2} - 2x - 24 = 0$

Medium

4. $3x^{2} - 19x + 6 = 0$
5. $4x^{2} + 3x - 1 = 0$
6. $2x^{2} - 16x + 30 = 0$

Hard

7. $x^{2} - 10x = -16$
8. $x^{2} + 8x = -12$
9. $x^{2} + 10x = -21$

Tips for teachers and parents

• Stay on Easy until students can find factor pairs quickly; the Medium and Hard levels assume that step is automatic.
• Use Medium to introduce the $a \cdot c$ (grouping) method, and ask students to check the fractional solution by substitution.
• Hard worksheets are a good habit check: before factoring anything, the student should rewrite each equation so one side is zero.
• Each worksheet prints with an answer sheet on a separate page. The Set ID on the worksheet matches the answer sheet, so you can print several different sets for one class.

Related topics

• Solving Equations Algebra
• Polynomial Multiplication Algebra 2
• Factoring Polynomials Algebra 2
• Functions Pre-Calculus