Question

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$(x + 1) ^{2} = 2(x - 3)$
Check whether the following are quadratic equations or not
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Answer 1

Answer:

$\: \boxed{\bf \: {(x + 1)}^{2} = 2(x - 3) \: is \: a \: quadratic \: equation \: }$

Step-by-step explanation:

Given equation is

$\sf \: {(x + 1)}^{2} = 2(x - 3)$

We know,

$\boxed{\sf \: {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \: }$

Using this algebraic identity, we get

$\sf \: {x}^{2} + {(1)}^{2} + 2(x)(1) = 2x - 6$

$\sf \: {x}^{2} + 1 + 2x = 2x - 6$

$\sf \: {x}^{2} + 1 + 2x - 2x + 6 = 0$

$\sf \: {x}^{2} + 7 = 0$

$\sf \: Since, \: it \: can \: be \: expressed \: as \: {ax}^{2} + bx + c = 0$

$\implies\sf \: \boxed{\sf \: {(x + 1)}^{2} = 2(x - 3) \: is \: a \: quadratic \: equation \: }$

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Additional Information:

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac

Answer 2

Yes, the equation (x+1)² = 2(x-3) is a quadratic equation.

A quadratic equation is an equation that can be written in the form of ax^2 + bx + c = 0, where x is the variable and a, b, and c are constants. In this case, we can simplify the equation as follows:

(x+1)² = 2(x-3)

x² + 2x + 1 = 2x - 6

x² + 2x - 2x + 1 + 6 = 0

x² + 7 = 0

As you can see, we have a quadratic expression of the form ax^2 + c = 0, where a = 1 and c = 7. Therefore, this equation is a quadratic equation.