How to Solve Quadratic Functions

A quadratic function describes a parabola. “Solving” usually means finding zeros (x-intercepts), the vertex, and key features of the graph so you can interpret or sketch it with confidence.

📉 Algebra⏱️ ~20 min read📊 Intermediate

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What You’ll Learn

✓How to recognize the three common forms of a quadratic function
✓How to find the vertex and axis of symmetry
✓How to solve for zeros (roots) and interpret what they mean
✓How a affects the parabola’s shape and direction

What Is a Quadratic Function?

A quadratic function has the form f(x) = ax^2 + bx + c with a ≠ 0. Its graph is a parabola. Depending on the problem, “solve the quadratic function” can mean different things:

Find the zeros (solutions to f(x) = 0), which are the x-intercepts.
Find the vertex (the maximum or minimum point).
Identify key features: axis of symmetry, opening direction, and y-intercept.

If your question is strictly about solving a quadratic equation, start with how to solve a quadratic equation step by step. This guide focuses on the function viewpoint (graph + features), while still showing how to compute roots.

The 3 Most Useful Forms

Standard form

ax^2 + bx + c

Best for y-intercept (c) and using x = −b/(2a) to find the vertex.

Factored form

a(x − r1)(x − r2)

Best for roots: the zeros are x = r1 and x = r2.

Vertex form

a(x − h)^2 + k

Best for vertex: the vertex is (h, k), and the axis is x = h.

How to Solve/Analyze a Quadratic Function (A Reliable Checklist)

Step 1: Identify a, b, c (or h, k)

If you have standard form ax^2 + bx + c, write down a, b, and c. If you have vertex form, write down h and k. This determines the fastest path to the vertex.

Step 2: Find the vertex and axis of symmetry

In standard form, the x-coordinate of the vertex is:

x_vertex = −b / (2a)

Then compute y_vertex = f(x_vertex). The axis of symmetry is the vertical line x = x_vertex.

Step 3: Solve for zeros (roots) by setting f(x) = 0

To find x-intercepts, solve ax^2 + bx + c = 0. You can factor, use square roots, complete the square, or use the quadratic formula.

If you need the quadratic formula specifically, see how to solve using the quadratic formula.

Step 4: Find the y-intercept and a few symmetric points

The y-intercept is f(0). For sketching, pick a point one unit to the right of the axis and one unit to the left (symmetry!). That gives you a fast, accurate shape.

Step 5: Interpret the parabola’s direction and width

If a > 0, the parabola opens up (vertex is a minimum). If a < 0, it opens down (vertex is a maximum). Larger |a| makes the parabola narrower; smaller |a| makes it wider.

Worked Example (Find Vertex + Roots)

Given: f(x) = x^2 − 4x − 5

1) Vertex

Here a = 1 and b = −4. So x_vertex = −(−4)/(2·1) = 2.

Now compute y_vertex = f(2): f(2) = 2^2 − 4·2 − 5 = 4 − 8 − 5 = −9.

Vertex is (2, −9) and the axis of symmetry is x = 2.

2) Roots (zeros)

Solve x^2 − 4x − 5 = 0. This factors: (x − 5)(x + 1) = 0.

So the zeros are x = 5 and x = −1. Those are the x-intercepts of the graph.

Common Mistakes

Mixing up vertex vs. roots

The vertex is where the parabola turns; roots are where it crosses the x-axis. They can be far apart.

Forgetting symmetry

Points the same distance left/right of the axis have the same y-value. Use symmetry to sketch faster.

Sign errors in −b/(2a)

If b is negative, −b becomes positive. Use parentheses: −(−4).

Assuming it always factors nicely

Many quadratics don’t factor over integers. In that case, use the quadratic formula or completing the square.

If you want more examples of how quadratics behave in different forms, Khan Academy’s quadratics lessons are a helpful companion.

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