Solve the equation (x-2)^2=49

\left(x-2\right)^2=49

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Answer

$x_1=9,\:x_2=-5$

Step by step solution

Problem

$\left(x-2\right)^2=49$
1

Expanding the polynomial

$4-4x+x^2=49$
2

Subtract $4$ from both sides of the equation

$x^2-4x=49-4$
3

Subtract the values $49$ and $-4$

$x^2-4x=45$
4

Rewrite the equation

$-45-4x+x^2=0$
5

To find the roots of a polynomial of the form $ax^2+bx+c$ we use the quadratic formula, where $a=1$, $b=-4$ and $c=-45$

$x =\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
6

Substituting the values of the coefficients of the equation in the quadratic formula

$x=\frac{-4\left(-1\right)\pm \sqrt{180+{\left(-4\right)}^2}}{2}$
7

Multiply $-1$ times $-4$

$x=\frac{4\pm \sqrt{180+{\left(-4\right)}^2}}{2}$
8

Calculate the power

$x=\frac{4\pm \sqrt{180+16}}{2}$
9

Add the values $16$ and $180$

$x=\frac{4\pm \sqrt{196}}{2}$
10

Calculate the power

$x=\frac{4\pm 14}{2}$
11

To obtain the two solutions, divide the equation in two equations, one when $\pm$ is positive ($+$), and another when $\pm$ is negative ($-$)

$x_1=\frac{4+ 14}{2}\:\:,\:\:x_2=\frac{4- 14}{2}$
12

Simplifying

$x_1=9,\:x_2=-5$
13

We found that the two real solutions of the equation are

$x_1=9,\:x_2=-5$

Answer

$x_1=9,\:x_2=-5$

Problem Analysis

Main topic:

Quadratic formula

Time to solve it:

0.24 seconds

Views:

75