Question

The number of the solutions of the equation $ {{5}^{2x-1}}+{{5}^{x+1}}=250, $ is/are

• A. $ 0 $
• B. $ 1 $
• C. $ 2 $
• D. $infinitely \,many$

Answer 1

The Correct Option is B

Given equation is $ {{5}^{2x-1}}+{{5}^{x+1}}=250 $
$ \Rightarrow $ $ {{5}^{2x}}{{.5}^{-1}}+{{5}^{x}}.5=250 $
$ \Rightarrow $ $ \frac{{{({{5}^{x}})}^{2}}}{5}+{{5}^{x}}.5=250 $ ?.(i) Let $ {{5}^{x}}=t $ Then E (i) becomes $ \frac{{{t}^{2}}}{5}+t.5=250 $
$ \Rightarrow $ $ {{t}^{2}}+25t=250\times 5 $
$ \Rightarrow $ $ {{t}^{2}}+25t-1250=0 $
$ \therefore $ $ t=\frac{-25\pm \sqrt{{{(25)}^{2}}-4\times 1\times (-1250)}}{2\times 1} $
$ \Rightarrow $ $ t=\frac{-25\pm \sqrt{625+5000}}{2} $
$ \Rightarrow $ $ t=\frac{-25\pm \sqrt{5625}}{2} $
$ \Rightarrow $ $ t=\frac{-25\pm 75}{2} $ On taking $ +ve $ sign, we get $ t={{5}^{x}}=\frac{-25+75}{2}=\frac{50}{2} $
$ \Rightarrow $ $ {{5}^{x}}=25 $
$ \Rightarrow $ $ {{5}^{x}}={{5}^{2}} $
$ \Rightarrow $ $ x=2 $ On taking $ -ve $ sign, we get $ t={{5}^{x}}=\frac{-25-75}{2} $ $ {{5}^{x}}=\frac{-100}{2} $ $ {{5}^{x}}=-50 $ Cannot express in terms of power of 5. Hence number of solution of given equation is 1.

Answer 2

Ans. 

• The word ‘quadratic equation’ is derived from the Latin word ‘quadratus’ which means ‘a square’.
• A quadratic equation is an equation with the form, ax2+bx+c = 0, where x represents an unknown, a, b, and c are constants, and a is not equal to 0.
• If a = 0, then the equation becomes linear, not quadratic.
• The constants a, b, and c are known as the coefficients.
• ‘c’ expresses the constant term, b is the linear coefficient, and ‘a’ is the quadratic coefficient.
• The quadratic equations include only one unknown variable and hence, are considered to be univariate.
• As the greatest power is two, they are known as second-degree polynomial equations.

The discriminant of a quadratic equation is described as the number D= b2- 4ac and is found from the coefficients of the equation ax2+bx+c = 0. The discriminant shows the nature of roots that an equation has.

b²- 4ac is found from the quadratic formula

Complex Number: Any number that is formed as a+ib is called a complex number. For example 9+3i, and 7+8i are complex numbers. Here i = -1. With this, we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.

Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.