Basics of Quadratic Equations | Mathematics (Maths) Class 11 - Commerce PDF Download

What are quadratic equations?

A quadratic equation is an equation with a variable to the second power as its highest power term. 

For example, in the quadratic equation 3x²- 5x-2=0 

• x is the variable, which represents a number whose value we don't know yet.
• '2' is the power or exponent. An exponent of 2 means the variable is multiplied by itself.
• '3' and '-5' are the coefficients, or constant multiples of x² and x. 3x² , -5x are single terms. 
• '-2' is a constant term.

How to Solve Quadratic Equations using Square Roots?

• Quadratic equations that lack x-terms, like $2x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}=$32, can be solved without needing to set the expression equal to zero.
• The first step is to isolate $x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}x^2$and then apply the square root to find x.
• When solving Quadratic equations by taking square roots, both the positive and negative square roots are considered solutions. because when we square a solution, the result is always positive.
• For example, in the equation $x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}=$4, both 2 and -2 are valid solutions:
• $2\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}=4$
• (−2$)\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}=4$
• To solve quadratic equations without x-terms:
• Step 1 : Isolate x²$x^2$
• Step 2 : Take the square root of both sides, remembering to include both positive and negative roots as solutions.

Example: What values of x satisfy the equation 2x² = 18?
Sol: To solve the equation $2x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}=182x^2\; =\; 18$ for x, follow these steps:

1. Isolate x²
Start by dividing both sides of the equation by 2 to isolate x²

$x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}=\frac{18}{2}x^2\; =\; \backslash frac\{18\}\{2\}$ x 2

2. Take the square root of both sides:
Now, apply the square root operation to both sides of the equation. Remember to consider both the positive and negative square roots:

x= ±√9 

x = ±3
The values of x that satisfy the equation $2x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}=18$ are $x=3$ and $\mathrm{x }$= -3.

Thus, the solutions are:

$x=3and\mathrm{ and\; x}$=−3$x\; =\; 3\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; x\; =\; -3$

Zero Product Property and Factored Quadratic Equations

• The zero product property states that if ab = 0, then either a or b is equal to 0.
• The zero product property lets us solve factored quadratic equations by solving two linear equations.
• For a quadratic equation such as (x-5)(x + 2) = 0, we know that either x -5 = 0 or x + 2 = 0. Solving these two linear equations gives us the two solutions to the quadratic equation.
• To solve the quadratic equation (x−5)(x+2)=0 using the zero product property following steps should be taken:

Step 1:  Set each factor to zero:

x - 5 = 0  → x = 5

x  + 2 = 0 → x = -2

Step 2: Solutions

x = 5

X = -2

Thus, the solutions are 𝑥 = 5 and  𝑥 = −2

Steps to solve a factored quadratic equation using the zero product property:

Step 1: Set each factor equal to 0.

Step 2:  Solve the equations by keeping variable on one side and constant on other. The solutions to the linear equations are also solutions to the quadratic equation.

How to Solve Factorable Quadratic Equations

• If we can write a quadratic expression as the product of two linear expressions (factors), then we can use those linear expressions to calculate the solutions to the quadratic equation.
• We'll focus on factorable quadratic equations with 1 as the coefficient of the x² term, such as x² - 2x - 3 = 0. For more advanced factoring techniques, including special factoring and factoring quadratic expressions with x² coefficients other than 1, check out the Factoring quadratic and polynomial expressions.
• The factors will be in the form (x + a)(x + b), where a and b fulfill the following criteria:
• The sum of a and b is equal to the coefficient of the x-term in the unfactored quadratic expression.
• The product of a and b is equal to the constant term of the unfactored quadratic expression.
• For example, we can solve the equation x² - 2x - 3 by factoring x²- 2x - 3 into (x + a)(x + b), where:
  • a + b is equal to the coefficient of the x-term, -2.
  • a * b is equal to the constant term, -3.
• For this equation, -3 and 1 would work as: 
  • -3 + 1 = -2
  • (-3)(1) = -3
• This means we can rewrite x² - 2x - 3 = 0 as (x - 3)(x + 1) = 0 and solve the quadratic equation using the zero product property. 
• Keep in mind that a and b are not themselves solutions to the quadratic equation! 
• When solving factorable quadratic equations in the form x² + bx + c = 0
• Steps to be Followed :

Step 1 :  Rewrite the quadratic expression as the product of two factors. The two factors are linear expressions with an x-term and a constant term. The sum of the constant terms is equal to b, and the product of the constant terms is equal to c. 

Step 2 :  Set each factor equal to 0.
Step 3 :  Solve the equations by isolating. The solutions to the linear equations are also solutions to the quadratic equation.

Quadratic formula

• Not all quadratic expressions are factorable, and not all factorable quadratic expressions are easy to factor.
• The quadratic formula gives us a way to solve any quadratic equation as long as we can plug the correct values into the formula and evaluate.
  
  

Steps to Solve Quadratic Equation using Quadratic Formula
To solve a quadratic equation using the quadratic formula:

• Step 1: Rewrite the equation in the form ax² + bx + c = 0.
• Step 2:  Substitute the values of a, b, and c into the quadratic formula, shown below.
  
• Step 3: Evaluate x.

The part of the quadratic formula under the square root, $b\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}-4acb^2\; -\; 4ac$, is called the Discriminant. The discriminant’s value determines the number of unique real solutions for the equation:

• If b²−4ac>0$b^2\; -\; 4ac\; >\; 0$, then $\surd$b2−4ac is a real number, so the quadratic equation has two distinct real solutions:

  

• If b²−4ac = 0$b^2\; -\; 4ac\; =\; 0$, then b²−4ac$\backslash sqrt\{b^2\; -\; 4ac\}$ equals $0$, simplifying the quadratic formula to:

  

  In this case, the equation has one unique real solution.

• If b²−4ac < 0, then √b²−4ac becomes imaginary, meaning the quadratic equation has no real solutions.

Example : What are the solutions to the equation  x2 - 6x = 9 \[x^2-6x=9\] ?

Solution:

Step 1: Rewrite the equation in standard form

The equation is:

$x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}-6x=9x^2\; -\; 6x\; =\; 9$

Move 9 to the left side to set the equation to zero:

$x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}-6x-9=0x^2\; -\; 6x\; -\; 9\; =\; 0$

Now, it’s in the form $ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">2</sup>}+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$ where:

• a=1
• b=−6
• c=−9

Step 2: Apply the Quadratic Formula

The quadratic formula is:

Substitute a=1, b=−6, and c=−9 :

Step 3: Simplify under the square root

Step 4: Simplify the square root

Step 5: Simplify the fraction

x = 3 ± 3√2

The solutions are:

$x=3+3\mathrm{\surd 2}and$ and x=3−3√2