How to Solve Quadratic Equations by Factoring
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 Made Simple
1. What is a quadratic expression and how is it different from a quadratic equation?
A quadratic expression is a polynomial of degree two, written in the form ax² + bx + c, where 'a' is not zero. It represents a value that changes as 'x' changes. A quadratic equation, on the other hand, sets this expression equal to zero: ax² + bx + c = 0. The key difference is that an expression is something you simplify or factor, while an equation is something you solve to find the specific values of 'x' (called roots) that make the statement true.
2. What is the general form of a quadratic equation as per the CBSE 2025-26 syllabus?
The general or standard form of a quadratic equation, as defined in the CBSE curriculum for the 2025-26 session, is ax² + bx + c = 0. In this form:
- 'x' is the variable.
- 'a', 'b', and 'c' are real numbers.
- The coefficient 'a' cannot be zero (a ≠ 0), because if it were, the x² term would disappear, and it would no longer be a quadratic equation.
3. What are the primary methods used to factor a quadratic expression?
There are several methods to factor a quadratic expression. The most common ones taught in the NCERT syllabus are:
- Splitting the Middle Term: This involves rewriting the middle term 'bx' as a sum or difference of two terms to group and factor the expression.
- Using Algebraic Identities: Applying standard identities like (a+b)², (a-b)², or a²-b².
- Completing the Square: This method transforms the expression into a perfect square trinomial, which is easy to factor.
- Using the Quadratic Formula: While this directly finds the roots, the roots can be used to write the factored form of the equation.
4. How do you factor a quadratic expression of the form ax² + bx + c by splitting the middle term?
To factor a quadratic expression by splitting the middle term, you follow a clear process. First, find the product of the coefficients 'a' and 'c'. Then, identify two numbers whose product equals 'ac' and whose sum equals 'b'. You can then rewrite the middle term 'bx' using these two numbers and proceed to factor the expression by grouping the terms.
5. When is it impossible to factor a quadratic expression using real numbers?
It is impossible to factor a quadratic expression using real numbers when its corresponding equation (ax² + bx + c = 0) has no real roots. This situation occurs when the value of the discriminant (D = b² - 4ac) is negative. A negative discriminant indicates that the graph of the quadratic does not intersect the x-axis, meaning there are no real-number solutions that can be used to form the factors.
6. How does the 'Zero Product Property' help in solving a factored quadratic equation?
The Zero Product Property is the key principle for finding roots after factoring. It states that if the product of two or more factors is zero, then at least one of those factors must be zero. For a factored quadratic like (px + q)(rx + s) = 0, this property allows you to set each factor equal to zero individually (px + q = 0 or rx + s = 0) and solve the resulting linear equations to find the roots of the original quadratic equation.
7. Can you provide a real-world example of where factoring quadratics is applied?
A common real-world application of factoring quadratics is in projectile motion. For instance, when a ball is thrown, its path creates a parabola that can be modelled by a quadratic equation. Factoring this equation helps determine key information, such as how long the ball is in the air or the time it takes to reach a certain height. This is crucial in fields like physics, sports analytics, and engineering.
8. What is the key difference when factoring a quadratic with a leading coefficient of 1 (a=1) versus a > 1?
The main difference lies in the complexity of the 'splitting the middle term' method.
- When a = 1 (e.g., x² + 5x + 6), you just need to find two numbers that multiply to 'c' (6) and add to 'b' (5). The factors are directly formed from these numbers: (x+2)(x+3).
- When a > 1 (e.g., 2x² + 7x + 3), you must find two numbers that multiply to the product of a × c (2 × 3 = 6) and add to 'b' (7). You must then use these numbers to split the middle term and perform an additional step of factoring by grouping to find the final factors.
