Solve the inequality 4x^2+1>4x

4x^2+1>4x

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Answer

$0.5>x>0.5$

Step by step solution

Problem

$4x^2+1>4x$
1

Moving the term $1$ to the other side of the inequation with opposite sign

$4x^2>4x-1$
2

Grouping terms

$4x^2-4x>-1$
3

Rewrite the inequation

$1-4x+4x^2>0$
4

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

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

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

$x=\frac{-4\left(-1\right)\pm \sqrt{{\left(-4\right)}^2-16}}{4\cdot 2}$
6

Multiply $-1$ times $-4$

$x=\frac{4\pm \sqrt{{\left(-4\right)}^2-16}}{8}$
7

Calculate the power

$x=\frac{4\pm \sqrt{16-16}}{8}$
8

Add the values $16$ and $-16$

$x=\frac{4\pm \sqrt{0}}{8}$
9

Calculate the power

$x=\frac{4\pm 0}{8}$
10

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

$x_1=\frac{4+ 0}{8}\:\:,\:\:x_2=\frac{4- 0}{8}$
11

Simplifying

$x_1=0.5,\:x_2=0.5$
12

Applying the quadratic formula we obtained the two solutions $x_1$ and $x_2$, with which we write the solution interval

$0.5>x>0.5$

Answer

$0.5>x>0.5$

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Problem Analysis

Main topic:

Quadratic formula

Time to solve it:

0.2 seconds

Views:

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