10a 2-39a 14 0 Use Factoring Calculator

This calculator helps you factor quadratic equations of the form ax² + bx + c = 0. Factoring is a method of solving quadratic equations by expressing the quadratic as a product of two binomials.

How to Use the Factoring Calculator

To use the factoring calculator:

Enter the coefficients for the quadratic equation in the form ax² + bx + c = 0
Click the "Calculate" button
View the factored form of the equation and the solutions
Use the chart to visualize the quadratic function

The calculator will show you the step-by-step process of factoring the quadratic equation and provide the solutions.

Factoring Formula

The general form of a quadratic equation is:

ax² + bx + c = 0

To factor this equation, you need to find two binomials (px + q) and (rx + s) such that:

(px + q)(rx + s) = ax² + bx + c

The factored form of the equation is:

(px + q)(rx + s) = 0

The solutions to the equation are the values of x that make either binomial equal to zero:

x = -q/p or x = -s/r

Worked Example

Let's factor the equation 10a² - 39a + 14 = 0:

Find two numbers that multiply to (10)(14) = 140 and add to -39
The numbers are -35 and -4 because (-35)(-4) = 140 and -35 + (-4) = -39
Rewrite the middle term using these numbers: 10a² - 35a - 4a + 14 = 0
Factor by grouping: (10a² - 35a) + (-4a + 14) = 0
Factor out common terms: 5a(2a - 7) - 2(2a - 7) = 0
Factor out the common binomial: (5a - 2)(2a - 7) = 0
Set each binomial equal to zero and solve: a = 2/5 or a = 7/2

The factored form of the equation is (5a - 2)(2a - 7) = 0, with solutions a = 0.4 and a = 3.5.

FAQ

What is the difference between factoring and completing the square?: Factoring is a method of solving quadratic equations by expressing the quadratic as a product of two binomials. Completing the square is another method that involves rewriting the quadratic in the form (x + p)² = q.
When should I use factoring instead of the quadratic formula?: Factoring is generally faster and simpler when the quadratic can be easily factored. The quadratic formula should be used when factoring is difficult or when the equation doesn't factor nicely.
What if the quadratic equation doesn't factor nicely?: If the quadratic doesn't factor nicely, you can use the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). This will give you the exact solutions to the equation.
Can the factoring calculator solve any quadratic equation?: The factoring calculator can solve quadratic equations of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. It works best when the equation can be factored easily.