How to Factor Quadratic Equations Step by Step with Examples
The equation of quadratic (from the Latin quadratus for "square") in algebra is an equation that can be rearranged in regular form as a standard form of a quadratic equation. In a quadratic equation, a variable is multiplied by itself, an operation known as squaring. This language comes from the area of a square multiplied by itself being its side length. The expression "quadratic" comes from quadratum, the word for the square in Latin. Many problems in physics and mathematics are in the form of quadratic equations. The solution of the quadratic equation is of special significance in mathematics. A quadratic equation, as already discussed, has no real solutions if D < 0. This case is of prime importance, as you can see in later lessons. This helps to establish a new area of mathematics called Complex Analysis.
The standard formula of quadratic is:
ax2 + bx + c = 0,
where x is an unknown number, and a, b, and c are known numbers, where a ≠ 0. If a = 0, the equation is linear, not quadratic.
Different Ways for Solving of Quadratic Equation:
Square Roots
Solve the quadratic equation ax2 + bx + c = 0 by completing the square. We know that a, b, and c are numbered here, but we have no idea what the values of all of them are. The only condition we know is, “a” cannot be zero.
First, because we do not want a coefficient on x2 as it increases the works, we divide both sides by a.
To get it out of the way, we then deduct c/a from both sides.
Next, we use b/a (x coefficient), split by 2, and square to find (b/2a)2.
This number is added on both sides.
Completing the Square
Divide all the terms by the value of a (the coefficient of x2).
Switch the number term (c/a) to the equation's right side.
On the left side of the equation, complete the square and offset this by applying the same value to the right side of the equation.
Take the square root of the equation on both sides.
To find x, deduct the number which remains on the left side of the equation.
Quadratic Formula
The quadratic formula is a formula in elementary algebra that provides the solution(s) to a quadratic equation. Instead of using the quadratic formula, there are other methods of solving a quadratic equation, such as factoring (direct factoring, grouping, AC method), completing the square, graphing, and others.
Where the plus-minus symbol "±" means that there are two solutions to the quadratic equation.
Steps to find the root of a quadratic equation:
By applying the values in the formula: \[x = \frac{-x \pm \sqrt{b^{2} - 4ac}}{2a}\]
There are few conditions to adhere to:
There is one real root while b2 - 4ac = 0 is present.
There are two real roots when b2 - 4ac > 0 is present.
There are two complex roots when b2 - 4ac < 0 is involved.
Factorisation Formula
There are three steps of factoring quadratic equations:
Check for two numbers that multiply to give ac (i.e. c times a), and add to give b.
With those numbers, rewrite the middle term.
Our two new terms should have a clearly identifiable common factor.
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Solved Examples
1. Write the solution of quadratic equation using factoring: x2 + 16 = 10x
Solution:
In the correct form, write the equation. With the terms written in descending order, we need to set the equation equal to zero in this case.
⇒ x2 - 10x + 16 = 0
To consider the problem, use a factoring technique.
⇒(x - 2)(x - 8) = 0
Set each factor containing a variable equal to zero by using the Zero Product Property.
⇒(x - 2) = 0 or (x - 8) = 0
By having the x on one side and the answer on the other, solve each factor that was set equal to zero.
Answer ⇒ x = 2 or x = 8
FAQs on Factoring Quadratics Methods and Explanation
1. What is factoring quadratics?
Factoring quadratics is the process of rewriting a quadratic expression in the form ax² + bx + c as a product of two binomials. A quadratic is factored into the form (px + q)(rx + s). This method is used to simplify expressions and solve quadratic equations by setting each factor equal to zero.
2. How do you factor a quadratic equation?
To factor a quadratic equation, rewrite it as a product of two binomials whose multiplication equals the original expression. For x² + bx + c (where a = 1):
- Find two numbers that multiply to c.
- Those numbers must add to b.
- Write the factors as (x + m)(x + n).
3. What is the AC method in factoring quadratics?
The AC method is a technique used to factor quadratics of the form ax² + bx + c when a ≠ 1. Steps:
- Multiply a × c.
- Find two numbers that multiply to ac and add to b.
- Split the middle term using those numbers.
- Factor by grouping.
4. How do you factor a quadratic when a is not 1?
When a ≠ 1, use the AC method or factoring by grouping. For example, to factor 3x² + 11x + 6:
- Multiply 3 × 6 = 18.
- Find numbers that multiply to 18 and add to 11 (9 and 2).
- Rewrite as 3x² + 9x + 2x + 6.
- Factor: (3x + 2)(x + 3).
5. What is the difference between factoring and the quadratic formula?
Factoring rewrites a quadratic as a product of binomials, while the quadratic formula directly calculates the solutions. The quadratic formula is x = (-b ± √(b² − 4ac)) / 2a. Factoring is quicker when the expression factors easily, but the quadratic formula works for all quadratic equations.
6. How do you factor a quadratic by grouping?
Factoring by grouping splits the middle term and factors common terms from pairs. Steps:
- Find two numbers that multiply to ac and add to b.
- Rewrite the middle term.
- Group the first two and last two terms.
- Factor out common factors.
7. Can all quadratic expressions be factored?
Not all quadratic expressions can be factored using integers. A quadratic can be factored over the real numbers only if its discriminant (b² − 4ac) is ≥ 0. If the discriminant is negative, the quadratic has complex roots and cannot be factored using real numbers.
8. How do you factor a perfect square trinomial?
A perfect square trinomial factors into the square of a binomial. It follows the pattern a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². Example: x² + 6x + 9 = (x + 3)².
9. How do you solve a quadratic equation by factoring?
To solve a quadratic by factoring, set the equation equal to zero and factor it, then use the zero product property. Steps:
- Write in standard form ax² + bx + c = 0.
- Factor the quadratic.
- Set each factor equal to zero.
- Solve for x.
10. What are common mistakes when factoring quadratics?
Common mistakes when factoring quadratics include choosing incorrect factor pairs and ignoring the greatest common factor (GCF). Key things to remember:
- Always factor out the GCF first.
- Check that factors multiply back to the original expression.
- Be careful with positive and negative signs.
- Ensure the expression is in standard form ax² + bx + c.
