Quadratic Formula Examples with Solutions

The quadratic formula is one of the most reliable tools in algebra. It solves any quadratic equation, even when factoring does not work. This guide covers the formula itself, explains the discriminant, and walks through multiple worked examples including word problems.

For a broader look at algebra fundamentals, see our Algebra Guide. If your equation can be factored easily, you may want to start with how to factor polynomials step by step.

The Quadratic Formula

For any equation in the form ax^2 + bx + c = 0, where a is not zero:

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

That is it. Identify a, b, and c, plug them in, and simplify. The +/- symbol means you will usually get two solutions — one using the plus and one using the minus.

The Discriminant

The expression under the square root, b^2 - 4ac, is called the discriminant. It tells you what kind of solutions to expect before you finish the calculation:

• b^2 - 4ac >0 — Two distinct real solutions.
• b^2 - 4ac = 0 — One repeated real solution (the parabola touches the x-axis at exactly one point).
• b^2 - 4ac <0 — Two complex (imaginary) solutions. No real x-intercepts.

Step-by-Step Process

1. Write the equation in standard form: ax^2 + bx + c = 0.
2. Identify a, b, and c.
3. Calculate the discriminant: b^2 - 4ac.
4. Plug into the formula and simplify.
5. Reduce fractions and simplify radicals if possible.

Want instant step-by-step solutions? Screenshot any quadratic equation with Math.Photos and get the full breakdown. Install free for Chrome.

Example 1: Two Real Solutions

Solve x^2 - 5x + 6 = 0

• a = 1, b = -5, c = 6
• Discriminant: (-5)^2 - 4(1)(6) = 25 - 24 = 1
• x = (5 +/- sqrt(1)) / 2
• x = (5 + 1) / 2 = 3 or x = (5 - 1) / 2 = 2

Solutions: x = 3 and x = 2

You can verify by factoring: x^2 - 5x + 6 = (x - 3)(x - 2). Both methods give the same answer.

Example 2: Two Irrational Solutions

Solve 2x^2 + 3x - 4 = 0

• a = 2, b = 3, c = -4
• Discriminant: 3^2 - 4(2)(-4) = 9 + 32 = 41
• x = (-3 +/- sqrt(41)) / 4

Since 41 is not a perfect square, the solutions stay in radical form:

Solutions: x = (-3 + sqrt(41)) / 4 and x = (-3 - sqrt(41)) / 4

As decimals, that is approximately x = 0.851 and x = -2.351.

Example 3: One Repeated Root

Solve 4x^2 - 12x + 9 = 0

• a = 4, b = -12, c = 9
• Discriminant: (-12)^2 - 4(4)(9) = 144 - 144 = 0

When the discriminant is zero, the +/- does not matter:

• x = (12 +/- 0) / 8 = 12/8 = 3/2

Solution: x = 3/2 (repeated)

This means the parabola just touches the x-axis at x = 3/2. You can confirm: 4x^2 - 12x + 9 = (2x - 3)^2.

Checking your work is just as important as solving. Use the free math checker to verify any quadratic solution.

Example 4: Complex (Imaginary) Solutions

Solve x^2 + 2x + 5 = 0

• a = 1, b = 2, c = 5
• Discriminant: 2^2 - 4(1)(5) = 4 - 20 = -16

A negative discriminant means complex solutions:

• x = (-2 +/- sqrt(-16)) / 2
• sqrt(-16) = 4i
• x = (-2 +/- 4i) / 2
• x = -1 +/- 2i

Solutions: x = -1 + 2i and x = -1 - 2i

Complex solutions always come in conjugate pairs. If you are in a course that does not cover imaginary numbers yet, a negative discriminant simply means “no real solutions.”

Example 5: Equation Not in Standard Form

Solve 3x^2 = 7x - 1

First, rearrange to standard form by moving everything to one side:

3x^2 - 7x + 1 = 0

• a = 3, b = -7, c = 1
• Discriminant: (-7)^2 - 4(3)(1) = 49 - 12 = 37
• x = (7 +/- sqrt(37)) / 6

Solutions: x = (7 + sqrt(37)) / 6 and x = (7 - sqrt(37)) / 6

Approximately x = 1.847 and x = 0.180.

Working through a full problem set? Math.Photos gives you step-by-step solutions for quadratics, factoring, and more. Try it free.

Example 6: Word Problem — Projectile Motion

A ball is thrown upward from a 48-foot platform with an initial velocity of 32 ft/s. Its height is modeled by h(t) = -16t^2 + 32t + 48. When does the ball hit the ground?

Set h(t) = 0:

-16t^2 + 32t + 48 = 0

Divide every term by -16 to simplify:

t^2 - 2t - 3 = 0

• a = 1, b = -2, c = -3
• Discriminant: (-2)^2 - 4(1)(-3) = 4 + 12 = 16
• t = (2 +/- sqrt(16)) / 2 = (2 +/- 4) / 2
• t = 3 or t = -1

Since time cannot be negative, the ball hits the ground at t = 3 seconds.

Example 7: Word Problem — Area

The length of a rectangle is 3 more than its width. The area is 70 square units. Find the dimensions.

Let w = width. Then length = w + 3.

w(w + 3) = 70

w^2 + 3w - 70 = 0

• a = 1, b = 3, c = -70
• Discriminant: 9 + 280 = 289
• w = (-3 +/- 17) / 2
• w = 7 or w = -10

Width must be positive, so w = 7 and length = 10.

The rectangle is 7 by 10 units.

Tips for Avoiding Mistakes

• Always rearrange to standard form (= 0) before identifying a, b, and c.
• Watch the sign of b carefully. If the equation is x^2 - 5x + 6 = 0, then b = -5, not 5.
• The denominator is 2a, not just 2. If a = 3, you divide by 6.
• Simplify radicals fully. For example, sqrt(12) = 2*sqrt(3).
• Do not forget to consider both the + and - cases unless the discriminant is zero.

FAQ

When should I use the quadratic formula instead of factoring?

Use factoring when the equation factors cleanly over the integers — it is faster. Use the quadratic formula when factoring is not obvious or when the discriminant is not a perfect square. The quadratic formula always works, so when in doubt, use it. See our factoring guide for more on when factoring is the better route.

What does it mean if the discriminant is zero?

It means the quadratic has exactly one solution (a repeated root). Graphically, the parabola just touches the x-axis at its vertex without crossing it.

Can the quadratic formula give me fractions?

Yes. Any time 2a does not divide evenly into -b +/- sqrt(discriminant), the result is a fraction. Always simplify the fraction at the end.

How do I handle complex solutions on a test?

If your course covers complex numbers, write the answer in a + bi form. If not, you can simply state “no real solutions” and note that the discriminant is negative.

For more algebra topics and techniques, visit our Algebra Guide.