Call

Quadratic Equations

A quadratic is any equation with an x² as its highest power. Learn every way to solve one — factoring, the square-root method, completing the square, and the quadratic formula — and how to read the parabola it graphs.

Start Your Free First Hour Algebra Tutoring
The Main Ideas

Four Ways to Solve, One Shape to Know

A quadratic is any equation whose highest power is \(x^2\). Every one graphs as a parabola, and every one can be solved — factoring is fastest when it works, and the quadratic formula never fails.

Standard form

\[ ax^2 + bx + c = 0, \quad a \neq 0 \]

Get everything on one side equal to zero first — every method below assumes that.

Method 1: Factoring

\[ x^2 - 5x + 6 = 0 \Rightarrow (x-2)(x-3) = 0 \]

If it factors, set each factor to zero (zero-product property): \(x = 2\) or \(x = 3\). Fast, but not everything factors nicely.

Method 2: Square-root method

\[ x^2 = k \Rightarrow x = \pm\sqrt{k} \]

When there is no plain \(x\) term, isolate the square and take the root of both sides. Do not forget the \(\pm\).

Method 3: Completing the square

\[ x^2 + bx = \left(x + \tfrac{b}{2}\right)^2 - \left(\tfrac{b}{2}\right)^2 \]

Rewrite as \((x + p)^2 = q\), then square-root. Always works, and it is where the quadratic formula comes from.

Method 4: The quadratic formula

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Plug \(a\), \(b\), \(c\) straight in. This always works — reach for it whenever factoring is not obvious.

The discriminant tells you how many roots

\[ D = b^2 - 4ac \]
  • \(D > 0\) → two real solutions
  • \(D = 0\) → one (repeated) solution
  • \(D < 0\) → no real solutions

Reading the parabola

vertex

Opens up when \(a > 0\), down when \(a < 0\). The vertex sits on the axis of symmetry at \(x = -\tfrac{b}{2a}\); the real solutions are where the parabola crosses the x-axis.

See It Worked Out

Worked Examples

Example 1 · Factoring

Solve: \(x^2 - 5x + 6 = 0\)

  1. Two numbers that multiply to \(+6\) and add to \(-5\): \(-2\) and \(-3\).

    \[ (x - 2)(x - 3) = 0 \]

  2. Set each factor to zero.

    \[ x - 2 = 0 \ \text{or}\ x - 3 = 0 \]
\( x = 2 \ \text{or}\ x = 3 \)

Example 2 · Square-root method

Solve: \(2x^2 - 50 = 0\)

  1. Isolate the square term.

    \[ 2x^2 = 50 \Rightarrow x^2 = 25 \]

  2. Take the square root of both sides — keep the \(\pm\).

    \[ x = \pm\sqrt{25} \]
\( x = 5 \ \text{or}\ x = -5 \)

Example 3 · Quadratic formula

Solve: \(x^2 + 4x + 1 = 0\)

  1. Identify \(a = 1\), \(b = 4\), \(c = 1\) and plug into the formula.

    \[ x = \frac{-4 \pm \sqrt{4^2 - 4(1)(1)}}{2(1)} \]

  2. Simplify under the root: \(16 - 4 = 12\), and \(\sqrt{12} = 2\sqrt{3}\).

    \[ x = \frac{-4 \pm 2\sqrt{3}}{2} \]

  3. Divide every term by 2.

    \[ x = -2 \pm \sqrt{3} \]
\( x = -2 + \sqrt{3} \ \text{or}\ x = -2 - \sqrt{3} \)
Your Turn

Practice Problems

Solve each equation, then reveal the solution to check yourself.

\(x^2 - 9 = 0\)

Show solution

Square-root method: \(x^2 = 9\).

\[ x = \pm 3 \]

\(x^2 + 7x + 10 = 0\)

Show solution

Factors: \(+2\) and \(+5\) multiply to 10 and add to 7.

\[ (x + 2)(x + 5) = 0 \Rightarrow x = -2 \ \text{or}\ x = -5 \]

\(x^2 - 6x + 9 = 0\)

Show solution

Perfect-square trinomial; the discriminant is 0, so one repeated root.

\[ (x - 3)^2 = 0 \Rightarrow x = 3 \]

\(3x^2 - 12 = 0\)

Show solution

Isolate: \(3x^2 = 12 \Rightarrow x^2 = 4\).

\[ x = \pm 2 \]

\(x^2 + 2x - 4 = 0\)

Show solution

Does not factor nicely — use the formula with \(a=1, b=2, c=-4\). Discriminant \(= 4 + 16 = 20\), \(\sqrt{20} = 2\sqrt{5}\).

\[ x = \frac{-2 \pm 2\sqrt{5}}{2} = -1 \pm \sqrt{5} \]
Keep Going

More Algebra Study Guides

FactoringSystems of Equations All Algebra guides →
Your Journey Starts Here

Ready to Help Your Student Succeed?

Meet your tutor first, no payment, no commitment. Just see the difference a great tutor makes.

First Hour Free Top 5% of Tutors Flexible Scheduling
★★★★★

“Your services have been a life saver! When my kids have a bad teacher, I have High Performance Tutoring. Thank you!”

Cynthia P., Salt Lake City Parent

We'll personally respond within one business day.

Claim Your Free First Session

Meet your tutor first, no payment required.

No payment required · We'll respond within 24 hours

Don't want to wait? Create your account and book your first session now →

Not sure where to start? Tell us about your student →