Quadratic Equations | StudyHelper Hub

Standard Form

ax² + bx + c = 0 (a ≠ 0)
Solutions are called roots or zeros — the x-values where the parabola crosses the x-axis.

Solving by Factoring

Factor the quadratic into two binomials, set each to zero.

x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = −2 or x = −3

The Quadratic Formula

x = (−b ± √(b²−4ac)) / 2a

2x² − 4x − 6 = 0: a=2, b=−4, c=−6
x = (4 ± √(16+48)) / 4 = (4 ± 8) / 4
x = 3 or x = −1

The Discriminant

b² − 4ac tells you the number of solutions:
> 0: two real solutions
= 0: one real solution (vertex touches x-axis)
< 0: no real solutions (parabola doesn't cross x-axis)

Completing the Square

This is the third way to solve a quadratic, and it is where the quadratic formula comes from. The idea is to force one side into a perfect square you can undo with a square root.

Solve x² + 6x − 7 = 0

Move the constant across: x² + 6x = 7
Take half of 6 (which is 3), square it (which is 9), and add it to both sides: x² + 6x + 9 = 16
The left side is now a perfect square: (x + 3)² = 16
Square root both sides, keeping both signs: x + 3 = ±4
Solve each: x = 1 or x = −7

Reading the Parabola

Every quadratic draws a parabola, and the numbers in the equation tell you its shape before you plot a single point.

If a > 0 the parabola opens upward and has a lowest point. If a < 0 it opens downward and has a highest point.
The larger |a| is, the narrower the curve.
The vertex sits at x = −b / 2a. Put that x back into the equation to get the y-coordinate.
The parabola is symmetric about the vertical line through its vertex, so the two roots are always the same distance from it.

Find the vertex of y = x² − 4x + 1

x = −(−4) / 2(1) = 2

y = 2² − 4(2) + 1 = −3

Vertex: (2, −3). Since a is positive, that is the minimum point of the curve.

Which Method Should You Use?

Factoring

Fastest when the numbers are friendly. Try it first on anything where a = 1 and you can spot two numbers that multiply to c and add to b.

Quadratic formula

Always works, every time. Use it the moment factoring stops being obvious, and on any problem with ugly decimals.

Completing the square

Best when you need the vertex, or when a problem asks for the equation in vertex form rather than just the roots.

Common Mistakes

Watch out for these

Forgetting to set the equation equal to zero before factoring. (x+2)(x+3) = 2 tells you nothing; only a product equal to zero does.
Losing the ±. A square root gives two answers, and dropping one costs you half the solution.
Sign errors on −b when b is already negative. If b = −4, then −b = +4.
Dividing only part of the numerator by 2a. The whole top, both terms, gets divided.

Where You Will Use This

Anything thrown or dropped follows a parabola, so quadratics predict how high a ball goes and when it lands.
Finding the maximum profit or minimum cost in a business problem means finding a vertex.
Area problems where length and width are related produce quadratics.
Satellite dishes and headlight reflectors are built as parabolas because of how they focus signals and light.

Summary

Standard form is ax² + bx + c = 0, and the solutions are where the parabola meets the x-axis.
Three solving methods: factoring, the quadratic formula, and completing the square.
The discriminant b² − 4ac tells you how many real solutions exist before you solve.
The vertex is at x = −b / 2a and gives the maximum or minimum value.

FAQ

Can every quadratic be factored? No — use the quadratic formula when it can't be factored nicely.

Why do quadratics usually have two answers? Because squaring destroys sign information. Both 3 and −3 square to 9, so undoing a square has to consider both possibilities.

What does it mean when the discriminant is negative? The parabola never touches the x-axis, so there is no real solution. The graph floats entirely above or entirely below the axis.

Do I have to check my answers? It is worth it. Substitute each root back into the original equation; both sides should come out equal. This catches sign slips immediately.

What is vertex form? y = a(x − h)² + k, where (h, k) is the vertex. Completing the square converts standard form into it.

Quick Quiz

Test what you just learned. Choose the best answer for each question.