Factoring Quadratics

What is Quadratic Equation?

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 (Frequently Asked Questions)

1. What is the General Form of the Quadratic Equation?

Ans: The general form of quadratic equation can be represented as ax2 + bx + c = 0.

A quadratic equation ax2 + bx + c = 0, may be expressed as a product (px + q)(rx + s) = 0. In certain situations, it is possible to evaluate, by simple observation, the p, q, r, and s values that make the two forms equal to each other. If the quadratic equation is written in the second form, the 'Zero Factor Property' states that if px + q = 0 or rx + s = 0, the quadratic equation is satisfied. The quadratic roots are given by solving these two linear equations.

2. What is the Formula of Quadratic Equation by Factoring?

Ans: There are three sections to the standard form of quadratic equations: ax2 + bx + c = 0, where a is the quadratic term coefficient, b is the linear term coefficient, and c is the constant. Using addition or subtraction, transfer both terms to one side of the equation, normally the left one. Factor out the whole equation. Set and solve any factor equal to zero. As a solution to the original equation, list each solution from the previous step. With the exception of special cases, such as where b = 0 or c = 0, inspection factoring only works for quadratic equations with rational roots. This means that it is difficult to solve the vast majority of quadratic equations that exist in practical applications by factoring through inspection.

3. What Professions Use the Quadratic Formula?

Ans: To explain the movement of objects that travel through the air, quadratic equations are also used. If you intend to join the army and work with cannons or tanks, then the quadratic equation will be used frequently to determine where shells will fall. These equations are used by engineers of all kinds. They are required, such as car bodies, for the design of any piece of equipment that is curved. They are also used by automotive engineers to design braking systems. Aerospace engineers work with them on a daily basis for similar reasons. With several complex structures involving quadratic equations, electrical and chemical engineers work. So do the engineers of machines. These equations are used by audio engineers to design sound systems that provide the highest possible quality of sound.