Quadratic Equation
Any equation is in the form of ax2+bx+c=0 where a=0 is a quadratic equation.
Q1: Check whether the following are quadratic equations:
(i) (x+1)2=2(x−3)
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Step-by-Step Solution
Step 1: Expand the equation
To determine if the given equation is a quadratic equation, we first need to expand both sides of the equation and simplify it. We will use the algebraic identity (a+b)2=a2+2ab+b2 for the left side and the distributive property for the right side.
Step 2: Simplify both sides
Applying the identity (x+1)2=x2+2(x)(1)+12=x2+2x+1 and distributing on the right side 2(x−3)=2x−6, we get the simplified form of the equation.
Step 3: Rearrange into standard form
Now, we move all terms from the right side of the equation to the left side to set the equation to zero. This allows us to combine like terms and express the equation in the standard quadratic form ax2+bx+c=0.
Step 4: Identify coefficients and conclude
The simplified equation is x2+7=0. Comparing this to the standard form of a quadratic equation, ax2+bx+c=0, we can identify the coefficients. Here, a=1, b=0, and c=7. Since the coefficient of x2 (which is a) is 1 (and not zero), the equation is indeed a quadratic equation.