Solve the equation -3x+5+x^2=8

x^2-3x+5=8

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Answer

$x_1=3.7913,\:x_2=-0.7913$

Step by step solution

Problem

$x^2-3x+5=8$
1

Subtract $5$ from both sides of the equation

$x^2-3x=8-5$
2

Subtract the values $8$ and $-5$

$x^2-3x=3$
3

Rewrite the equation

$-3-3x+x^2=0$
4

To find the roots of a polynomial of the form $ax^2+bx+c$ we use the quadratic formula, where $a=1$, $b=-3$ and $c=-3$

$x =\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
5

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

$x=\frac{-3\left(-1\right)\pm \sqrt{12+{\left(-3\right)}^2}}{2}$
6

Multiply $-1$ times $-3$

$x=\frac{3\pm \sqrt{12+{\left(-3\right)}^2}}{2}$
7

Calculate the power

$x=\frac{3\pm \sqrt{12+9}}{2}$
8

Add the values $9$ and $12$

$x=\frac{3\pm \sqrt{21}}{2}$
9

Calculate the power

$x=\frac{3\pm \sqrt{21}}{2}$
10

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

$x_1=\frac{3+ \sqrt{21}}{2}\:\:,\:\:x_2=\frac{3- \sqrt{21}}{2}$
11

Simplifying

$x_1=3.7913,\:x_2=-0.7913$
12

We found that the two real solutions of the equation are

$x_1=3.7913,\:x_2=-0.7913$

Answer

$x_1=3.7913,\:x_2=-0.7913$

Problem Analysis

Main topic:

Quadratic formula

Time to solve it:

0.22 seconds

Views:

75