# Quadratic Equation B1

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https://algebrica.org/exercises/quadratic-equation-b-1/
Fetched from algebrica.org test 4487; source modified 2025-03-06T18:03:26.

Solve the [quadratic equations](../quadratic-equations.md) using the [factorization method](../factoring-quadratic-equations.md).

\\[
x^2-5x + 6 = 0
\\]

* * *

The equations is in the standard form \\(ax^2+bx+c=0\\). First, it is essential to verify the its [discriminant](../quadratic-formula.md) \\(\\Delta = b^2 - 4ac\\) is \\(\\geq0\\) to ensure the equation admits solutions in the field of real numbers. Substituting the coefficients of the equation into \\(\\Delta\\), we get:

\\[
\\Delta =(-5)^2-4(1)(6) = 1 \\gt 0
\\]

\\(\\Delta \\gt 0 \\) means the equation has real solutions.

* * *

To find the equation’s solutions using the [factorization method](../factoring-quadratic-equations.md), we must find two numbers whose sum equals \\(b\\), or \\(-5\\) in this case, and whose product is equal to \\(a \\cdot c\\), or \\(6\\). It is immediate to see that the polynomial \\(x^2-5x + 6\\) is factorizable as the product of two [binomials](../binomials.md) \\((x-2)\\) and \\((x-3)\\).

We obtain:

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

* * *

The values of \\(x\\) that make the product null of \\((x-2)(x-3)\\) are \\(x=2 \\) and \\(x=3\\).

The solution to the equation is:

\\[
x\_1= 2, \\ x\_2 =3
\\]

Remember that the [discriminant](../quadratic-formula.md) is crucial in determining the nature and number of solutions of quadratic equations.

-   If \\( b^2 - 4ac > 0\\), the quadratic equation has two distinct real solutions.

\\[
S = \\{x\_1, x\_2\\} \\quad x\_1, x\_2 \\in \\mathbb{R} \\quad x\_1 \\neq x\_2
\\]


-   If \\( b^2 - 4ac = 0\\), the quadratic equation has two coincident real solutions.

\\[
S = \\{x\\} \\quad x \\in \\mathbb{R} \\quad x = x\_1 = x\_2
\\]


-   If \\( b^2 - 4ac = < 0\\), the quadratic equation has no real solutions. Instead, it gives rise to complex solutions characterized by imaginary components.

\\[
\\nexists \\hspace{10px} x \\in \\mathbb{R}
\\]