Chapter 4, Problem 23AR

To determine

To solve the given equation, $x^2-5x+3=0$ by factoring or quadratic formula.

Answer to Problem 23AR

  $x=\frac{5+\sqrt{13}}{2},\frac{5-\sqrt{13}}{2}$

Explanation of Solution

Given:

The equation is, $x^2-5x+3=0$

Calculation:

We can solve the given equation by using quadratic formula.

That is,

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The given equation $x^2-5x+3=0$ , which is in the form of $a{x}^2+bx+c=0$ . Where, $a=1,b=-5\text{ and c =3}$

Now, Substitute the values in the formula then we get that,

  $\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-\left(-5\right)\pm \sqrt{{\left(-5\right)}^2-4\left(1\right)\left(3\right)}}{2\left(1\right)}\\ x=\frac{5\pm \sqrt{25-12}}{2}\\ x=\frac{5\pm \sqrt{13}}{2}\\ x=\frac{5+\sqrt{13}}{2},\frac{5-\sqrt{13}}{2}\end{array}$

Therefore, the roots for the given equation, $x^2-5x+3=0$ are $x=\frac{5+\sqrt{13}}{2},\frac{5-\sqrt{13}}{2}$ .